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THE  UNIVERSITY 
OF  ILLINOIS 
LIBRARY 
62.1.8 

L57n 


Return  this  book  on  or  before  the 
Latest  Date  stamped  below. 


University  of  Illinois  Library 


THE  UNIVERSITY 


OF  ILLINOIS 


LIBRARY 

&2.I.8 

L57n 


Mimeographed 

by 

Edwards  Brosc 
Ann  Arbor, 
Mich, 


II  0 T E S 


on 


ELEMENTARY  M ' A 0 H I IT  E DESIGN 


Frepared  for  Student 


in  the 


Mechanical  Engineering  Department 


0.  A.  Leutwiler 
and 

W . V . Dunk in „ 

University  of  Illinois 
September , 1911 . 

Published  by 


The  U3  of  Ic  Supply  Store 
Oharapa i rn , 111. 


Digitized  by  the  Internet  Archive 
in  2017  with  funding  from 

University  of  Illinois  Urbana-Champaign  Alternates 


https://archive.org/details/notesonelementarOOIeut 


CHAPTER  I 


Materials  Used  in  the  Construct i on 
of  Machine  Parts c 

The  principal  materials  used  in  the  construction  of 
machine  parts  are  cast  iron,  steel,  wrought  iron,  copper,  trass, 
bronze,  babbitt  metal,  wood  and  leather* 

1.  Cast  iron  is  more  commonly  used  than  any  other  material 
in  machine  parts*  This  is  because  (a)  of  its  high  compressive 
strength  and  (t)  because  it  can  be  given  easily  any  desired  form* 

A wood  pattern  (sometimes  metal)  of  the  piece  desired  is  made; 

and  from  this  a mold  is  made  in  the  sand*  The  pattern  is  next, 

removed  from  the  mold  and  the  liquid  metal  poured  in,  which  on 
cooling  assumes  the  form  of  the  pattern* 

Cast  iron  is  obtained  directly  from  the  melting  of  the 
iron  ore  in  the  blast  furnace.  This  product  from  the  blast  furnace 
is  known  commercially  as  pig  iron*  It  fuses  easily  (melting  point 
PP00°  F. ) but  it  cannot  be  tempered,  or  welded  under  ordinary 
conditions.  The  composition  of  cast  iron  varies  considerably  but 
in  general  is  as  follows’ 

Metallic  iron *.90*0  to  95.0$ 

Garb cn . 1.5  " 4 . 5$ 

Till  icon  . 0.5  " 4.0  $ 

less  than  0.15$. 


Sulphur 


192 


- 


' 


' 


’ 

■ • • 


Phosphorus 

Manganese 


a e 


. 0 9 06  to  I o 5$ 

. trace  to  5,0^, 

Carbon  may  be  united  either  chemically  with  the  iron, 
in  which  case  the  product  is  known  as  white  iron,  or  it  may  exist 
in  the  free  state  when  the  product  is  known  as  grey  iron.  The 
white  iron  is  very  brittle  and  hard,  and  is  therefore  but  little 
used  in  machine  parts . In  the  free  state  the  carbon  is  known  as 
graphite . 

Silicon  is  an  important  constituent  of  cast  iron  because 
of  the  influence  it  exerts  on  the  condition  of  the  carbon  present 
in  the  iron,  From  0,25$  to  of  silicon  tends  to  wake  the 

iron  soft  and  strong*  but  beyond  2,0  ' silicon  the  iron  becomes 
weak  and  hard.  An  increase  of  silicon  causes  less  shrinkage  in  the 
castings,  but  a further  increase  (above  5%)  may  cause  an  increase 
in  the  shrinkage,  kith  about  lt0';4  silicon  the  tendency  to  pro- 
duce blowholes  in  the  castings  is  reduced  to  a minimum. 

Sulphur  in  cast  iron  causes  the  carbon  to  unite 
chemically  with  the  iron,  thus  producing  hard,  white  iron,  which  is 
hard  and  brittle.  For  good  castings,  the  sulphur  content  should 
not  exceed  0.15//. 

Phosphorus  in  cast  iron  tends  to  produce  weak  and. 
brittle  castings.  It  also  causes  the  metal  to  bo  very  fluid  when 
melted,  thus  causing  the  metal  to  take  an  excellent  impression  of 
the  mold.  Phosphorus  is,  therefore,  a desirable  constituent  in 
pig  iron  for  the  production  of  fine  thin  castings  where  no  great 
strength  is  required.  For  this  purpose,  from  5 to  5 " of  the  ol em- 
inent may  be  used.  The  absence  of  phosphorus  from  the  iron  give-’ 

rise  to  soft  and  malleable  castings,  which  are  also  lacking 

192 


- 


I 


f 


' 


V- 


3 


in  strength  and  soundness . For  strong  castings  of  good  quality, 
the  amount  of  phosphorus  rarely  exceeds  n.55$,  hut  when  fluidity 
and  softness  is  more  important  than  strength,  from  ifo  to  1,5$  may 
he  used. 

Manganese  when  present  in  cast,  iron  up  to  about  1.5'* 
tends  to  make  the  castings  harder  to  machine;  but  renders  them 
more  suitable  for  smooth  or  polished  surfaces.  It  also  causes  a 
fine  granular  structure  in  the  castings  and  prevents  the  absorption 
of  the  sulphur  during  melting.  Manganese  may  also  be  added  to 
cast  iron  to  soften  the  metal.  This  is  due  to  the  fact  that  the 
manganese  counteracts  the  effects  of  the  sulphur  and  silicon  by 
eliminating  the  former  amd  counteracting  the  latter.  However,  when 
the  iron  is  remelted,  its  hardness  returns  since  the  manganese  is 
oxidised  and  more  sulphur  is  absorbed.  The  transverse  strength  of 

oast  iron  nay  be  increased  about  30.0$,  and  the  shrinkage  and 
depth  of  chill  decreased  25o0$,  while  the  combined  carbon  is 
diminished  one-half  by  adding  to  the  molten  metal,  powdered  ferro- 
manganese in  the  proportion  of  one  pound  of  the  latter  to  about 
300  pounds  of  the  former. 

» Pig  inon  is  the  basis  for  the  manufacture  of  all  iron 

products.  It  is,  not  only  used  practically  unchanged  to  produce 

castings  of  a great  variety  of  form  and  quality,  but  it  is  also 

used  in  the  manufacture  of  wrought  iron  and  steel.  For  each  special 

purpose,  the  iron  must  have  composition  within  certain  limits.  It 

follows,  therefore,  that  pig  iron  offers  a considerable  variety 

of  composition.  The  practice  of  purchasing  pig  iron  by  analysis  is 

generally  followed  at  the  present  time.  The  following  table 

192 


. 


. 


. 


. 


■ 


■■ 


gives  the  specifications  as 

required 

by  one  large  manufacturing 

concern.  (I.  A 

p 

u 

% 

l — 1 

o 

> 

• 

p 829). 

Table  I 

0 

Class  Silicon 

Phosphorus 

Mangane  s 

Sulphur 

Total  Carbon 

* 

% 

Hot  over 
/° 

Hot  over 

i 

Hot  under 
<4 

1 1.5  to  8.0 

0e2  to  0.75 

1.0 

0.040 

5 .0 

2 2.0  to  0(5 

0,2  to  0.75 

1.0 

0 .035 

r *5 

S 2.5  to  3,0 

0.2  to  0.75 

1 .0 

0.050 

O c c 

4 2.0  to  2.5 

1.0  to  1.50 

1.0 

0 . 040 

37  K 

5 4.0  to  5,0 

0,2  to  0.80 

1.0 

0 . 040 

3v0 

Analysis  i 

s made  from 

drillings 

from  a pig 

selected  at 

random  from  each  four  tons  of  every  carload  as  unloaded.  The 


right  is  reserved  to  reject  a portion  or  all  of  the  material 
which.  does  not  conform  to  above  specifications  in  every  particular. 
In  a general  way,  the  specified  limits  for  the  composition 
of  the  chief  grades  of  pig  iron  are  as  follows: 


Grade  of  Iron 

Silicon 

Table  II. 

Sulphur 

Phosphorus 

Manganese 

Ho.  1 Foundry 

7®  /3 

2.5  to  3.0  Under 

.055 

c 

o 

r~* 

o 

SSL-P 

LO 

o 

! 

Under 

1 .0 

Ho . 2 " 

2.0  to  2.5 

ti 

.045 

0.5  to  1 ,00 

it 

1 .0 

No . 3 ,r 

1.5  to  2.0 

(t 

.055 

0.5  to  1.00 

it 

1 .0 

Malleable 

0.7  to  1.5 

ft 

,050 

Under  0,20 

it 

1,0 

Gray  Forge 

Und_er  1.5^ 

II 

.100 

" 1,00 

ii 

1 ,0 

Bessemer 

1.0  to  2.0 

II 

,050 

0.10 

it 

1 .0 

Low  Phosphorus 

Under  2.0 

II 

j 0 5 0 

” 0 , 30 

ii 

1 .0 

Basic 

M 1.0 

tl 

*■050 

n 1.00 

ii 

1.0 

Basic  Bessemer 

” 1.0 

If 

.050 

2,0  to  5,00 

1.0  to 

2.0 

According 

to  U30,  pig 

iron 

may  be 

separated 

roughly 

into 

tvr  o great  clan 

ses.  The  first  cla 

ss  includes  those 

grades 

used 

192 


r 

* 


. 


4 

I 


l ' 


1 


5 


the  production  of  foundry  and  malleable  irons,  while  the  second 
includes  those  used  in  the  manufacture  of  wrought  iron  and  steel. 
In  the  process  of  remelting  or  manufacturing,  the  first  class 
undergo  little  if  any  chemical  change,  while  the  second  class 
undergo  a complete  chemical  changer 

2.  Malleable  Castings  are  made  by  thoroughly  cleaning 
and  heating  foundry  castings  (preferably  with  the  sulphur  content 
low)  in  an  annealing  furnace  in  connection  with  some  substance 
that  will  absorb  the  carbon  from  the  cast  iron.  Hematite  or 
brown  iron  ore  in  pulverised  form  is  used  extensively.  The  in- 
tensity of  heat  required  is,  on  the  average,  about  1650°  F.  The 
length  of  time  the  castings  remain  in  the  furnace  depends  upon 
the  degree  of  maleability  required  and  upon  the  size.  Usually  the 
light  castings  require  a minamum  of  cO  hours  while  the  heavier 
ones  may  require  72  hours  or  longer. 

3.  Chilled  castings  are  those  which  have  a hard  and. 
durable  surface.  The  iron  used  is,  generally,  close  grained  gray 
iron,  low  in  silicon.  A chilled  casting  is  formed  by  making  that 
part  of  the  mold  in  contact  with  the  surface  of  the  casting  to  be 
chilled,  of  a construction,  ouch  that  the  heat  will  be  rapidly 
withdrawn.  The  mold  for  causing  the  chill  usually  consists  of 
iron  bars  or  plates,  placed  such  that  their  surfaces  will  be  in 
contact  with  the  molten  iron.  These  plates  abstract  heat  rapid- 
ly from  the  iron,  with  the  result  that  the  part  of  the  casting 

in  contact  with  the  cold  surface  assumes  a state  similar  to  white 
iron,  while  the  rest  of  the  casting  remains  in  tho  for?"  of  gray 

1 Oo 


. 


' 


. 


6 


iron.  The  withdrawal  of  heat  is  hastened  by  the  circulation  of 
cold  water  through  pipes,  circular  or  rectangular  in  cross  section, 
placed  near  the  surface  to  he  chilled.  Chilled  castings  offer  great 
resistance  to  crushing  forces.  The  outsid.e,  or  "shin"  of  the  ordi- 
nary casting  is  in  fact  a chilled  surface,  hut  by  the  arrangement 
mentioned  above  the  depth  of  the  "shin"  is  greatly  increased  with 
a corresponding  increase  in  strength  and  wearing  qualities.  Car 
wheels  and  chilled  rolls  are  familiar  examples  of  chilled  castings. 
Car  wheels  require  great  strength  combinedrwith  a hard  durable 
tread.  The  depth  of  chill  varies  from  3/8"  to  1",  (I.A.Vol.76,  p 

162).  Chilled  rolls  are  used  in  rolling  steel  and  iron  sheets,  and 
also  tin  plates,  because  their  hard  smooth  surfa.ce  gives  to  the 
sheets  and  plates  a smooth  surface,  {I. A.,  1903,  Apr .23,  p 2.) 

4.  Wrought  iron  is  formed  from  pig  iron  by  melting  the 
latter  in  a "puddling  furnace" t During  the  process  of  melting,  the 
impurities  in  the  pig  iron  are  removed  by  oxidation  leaving  the  pure 
iron  and"slag",  both  is  a pasty  condition.  In  this  condition 
the  iron  and  slag  is  formed  into  "muck  balls",  weighing  about  130 
pounds,  and  removed  from  the  furnace.  These  balls  are  put  into  a 
"squeezer"  and  compressed,  thereby  removing  a large  amount  of  the 
slag,  and  then  rolled  into  bars.  The  bars,  known  as  "muck  bars", 
are  cut  into  smaller  bars  or  strips  and  arranged  in  piles,  the  con- 
secutive layers  being  at  right  angles  to  each  other,  These  piles 
are  raised  to  a welding  heat  and  rolled  into  "merchant  bars" . 

These  bars  are  the  ordinary  wrought  iron  of  commerce.  Wrought  iron 
is  not  so  extensively  used  now  as  formerly,  steel,  to  a great  extent 

having  taken  its  place.  Wrought  iron,  however,  still  finds 

192 


' 


« 


_ 


7 

extensive  use  in  the  manufacture  of  pine,  formings,  parts  of  elec- 
trical machinery,  small  structural  shape"’  arv.i  crucible  steel. 

5.  Steel  is  a compound  in  which  iron  and  carbon  are 
the  principal  parts.  It  is  made  from  pig  iron  by  burning  out  the 
carbon,  silicon,  manganese  and.  other  impurities,  and  recarbonising 
to  any  degree  desired.  The  principal  processes  or  methods  of 
manufacturing  steel  are  (l)  the  Bessemer,  (2)  the  Open  Hearth  and 
( o ) the  Cement at i on  * 

Bessemer  Process.  - In  the  Bessemer  process,  several 
tons  (usually  about  ten)  of  molten  pig  iron  are  poured  into  a pear 
shaped  vessel  called  a converter.  Through  this  mass  of  iron  large 
quantities  of  cold  air  are  passed.  In  about  four  minutes,  all  the 
silicon  and  manganese  of  the  pig  iron  has  combined  with  the  oxygen 
of  the  air.  The  carbon  in  the  pig  iron  noi;r  begins  to  unite  with 
the  oxygen  forming  carbon-monoxide,  which,  burns  out  through  the 
mouth  of  the  converter  in  a long  brilliant  flame.  The  burning,  of 
the  carbon-monoxide  continues  for  about  six  minuted,  when  the  flame 
shortens,  thus  indicating  that  nearly  all  the  carbon  has  boon 
burned  out  of  the  iron,  and  that  the  air  supply  should  ho  shut  off. 
The  burning  out  of  these  impurities  has  raised  the  temperature  of 
the  iron  to  white  heat  and  left  a relatively  pure  mass  of  iron. 

To  this  mass  is  added  a certain  amount  of  carbon  in  the  form  of  a 
very  pure  iron  high  in  carbon  and  manganese.  The  metal  in  then 
poured  into  molds  forming  ingots,  which  aro  tolled  while  hot  into 
the  desired  shapes. 

The  characteristics  of  the  Bessemer  process  are: 

192 


8 


(l)  great  rapidity  of  reduction,  about  ton  minutes  perjb.eat;  (8) 
no  extra  fuel  required;  (f>)  metal  is  not  molted  in  the  furnace 
where  the  reduction  takes  place, 

Bessemer  steel  was  formerly  used  almost  entirely  in  the 
manufature  of  wire,  skelps  for  tubing,  wire  nails,  shafting, 
machine  steel,  tank  plates,  and  structural  shapes.  Open  hearth 
steel  has,  however,  very  largely  superseded  the  Bessemer  products 
in  the  manufacture  of  these  articles. 

Open  Hearth  Process . In  the  manufacture  of  open  hearth 
steol,  the  molten  pig  iron  direct  from  the  reducing  furnace,  is 
poured  into  a long  hearth,  the  top  of  which  has  a fire  brick  lin- 
ing. The  impurities  in  the  iron  are  burned  by  heat  reflected  from 
this  refractory  lining,  and  obtained  from  burning  gas  and  air. 

The  slag  is  first  burned,  and  the  slag  in  turn  oxidises  the  im- 
purities. The  time  required  for  purifying  is  from  6 to  10  hours, 
after  which  the  metal  is  recarbonized,  cast  into  ingots  and 
rolled  as  in  the  Bessemer  process. 

The  characteristics  of  the  open  hearth  process  are:  (1) 
relatively  long  time  to  oxidize  impurities;  (8)  large  quantities 
(85  to  70  tons)  purified  and  recarbonized  in  one  charge;  (5)  ex- 
tra fuel  (gas)  required;  (4)  a part  of  the  charge  (steel  scrap  and 
iron  ore,  added  to  the  charge  at  the  beginning  of  the  process)  are 
melted  in  the  furnace. 

Open  hearth  steel  in  used  in  the  manufacture  of  cutlery, 
files,  shovels,  picks,  boiler  plato,  and  armor  plate,  in  addition 
to  the  articles  mentioned  above. 

Cementation  Process,  In  this  process  of  manufacturing 

192 


. 


/ 


o 


steel,  bars  of  wrought  iron,  imbedded  in  charcoal  are  hostel  for 
several  clays.  The  wrought  iron  absorbs  carbon  fron  th s ?•••»  srcooJ. 
and  is  thus  transformed  into  steel.  Y?hen  the  bars  of  iron  c,r ' re- 
moved they  aro  found  to  be  covered  with  scales  or  "blisters'.  Tho 
name  given  to  this  product  is  "blister  steel"  , By  removing  the 
sclaes  and  blisters  and  subjecting  the  bars  to  a cherry  red  heat  for 
a few  days,  a more  uniform  distribution  of  the  carbon  is  obtained. 

Blister  steel,  when  heated  and  rolled  directly  into  the 
finished  bars,  is  known  as  German  Steel. 

Bars  of  blister  steel  may  be  cut  up  and  forged  together 
under  the  hammer,  forming  a product  called  "s'  oar  steel".  By  re- 
peating the  process  with  the  shear  steel,  we  obtain  "double  s? oar" 
steel . 

Crucible  Steel.  Crucible  or  cast  steel  is  very  uniform 
and  homogenous  in  structure.  It  in  made  by  molting  blister  s.t'/ul 
in  a crucible,  casting  it  in  ingots  and  rolling  into  bars.  By  this 
method  is  produced  the  finest  crucible  or*  cast  steels.  Another 
method  of  producing  crucible-cast  steel  is  to  melt  Swedish  iron, 
(wrought  iron  obtained  from  the  reduction  of  a very  pure  iron  ore 
in  the  blast  furnace,  and  in  using  charcoal  instead  of  coke  in  the 
puddling  flame)  and  charcoal  in  a sealed  vessel,  tho  contents  of 
which  are  poured  into  a larger  vessel  or  ladle,  containing  a simi- 
lar product  from  other  sealed  vessels.  The  metal  in  this  larger 
vessel  is  cast  into  ingots,  which  are  tubsequontly  forged  or  roll- 
ed. into  bars.  By  far  the  greater  portion  of  cast  stel  is  produced 
by  this  method. 


192 


. 


• 

' 

* 


i • ' 

1 

10 


6.  Steel  Castings,  Castings,  similar  to  cast  iron  cast- 
ings, may  be  formed  in  almost  any  desired  stare  from  too  molten 
steel.  The  open  hearth  steel  is  considered  surer i or  to  Bessemer 
steel  for  steel  castings.  In  texture,  these  castings  arc  course  and 
crystalline,  since  the  steel  has  been  allowed  to  cool  without 
drawing  or  rolling.  Formerly,  trouble  was  experienced  in  obtaining 
good,  sound  castings;  but  by  great  care  and  irriproved  methods  in 

the  production  of  molds,  first  class  castings  may  now  be  obtained. 
Steel  castings  are  used  for  those  machine  parts,  requiring  greater 
strength  than  is  obtained  by  using  pig  iron  castings, 

7.  Cold  Rolled  Steel,  This  steel  is  rolled  hot  to 
approximately  the  required  dimensions.  The  surface  is  then  care- 
fully cleaned,  usually  by  chemical  means,  and  rolled  cold  to  a 
very  accurately  gauged  thickness  between  smooth  rollers.  A very 
smooth  and  hard  surface,  with  greatly  increased  strength,  is  thus 
given  to  the  steel.  Cold  rolled  steel  is  use’  chiefly  for  shafting. 

8.  Special  Steels.  A special  steel  is  one  in  vrhicli  some 
other  element  rather  than  carbon  distinguishes  the  metal.  The 
principal  elements  used  to  form  these  special  steels  are  nickel, 
tungsten,  chromium,  manganese,  vanadium,  aluminium  and  silicon,. 

ITickel  steel  contained  0.0$  to  1.0;!  carbon,  and  up  to  >5$ 

nickel.  It  possesses  great  tensile  strength  and  ductility.  Tests 

show  that  its  tensile  strength  varies  from  100,000  to  275,000 

pounds  per  square  inch,  with  an  elastic  limit  of  40,000  to  75,000 

pounds  per  square  inch.  The  effect  of  the  nickel  is  not  always 

uniform;  thus  a nickel  coontent  up  to  8$  increases  the  tensile 

strength  and  elastic  limit,  while  between  8$  and  15$  brittleness  is 

192 


■ 


* 


. . ■ 


' 


' 


11 


produced;  but  above  15$  the  strength  and  elastic  Unit  return. 

Nickel  in  used  in  ordnance  work  and  in  making  am. or  "slate.  It 
does  not,  in  general,  crack  when  pierced  by  a projectile.  R .its 
made  from  this  steel,  show  better  wearing  qualities  than  tbos \*de 
from  Bessemer  steel.  On  account  of  its  ability  to  withstand  heavy 
shocks  and  torsional  stresses,  nicket  steel  is  used  for  crank  shafts 
shafting,  connecting  rods,  explosive  engines,  and  in  automobile 


work . 


Tungsten  steel  is  an  alloy  of  iron,  carbon,  tungsten, 
and  manganese,  and  sometimes  chromium.  The  element  which  gives 
this  steel  its  peculiar:  property  - self  or  air  hardening  - is  not 
tungsten  but  manganese  combined  with  carbon.  The  s tungsten, 
however,  is  an  important  element,  since  it  enables  the  alloy  to 
contain  a larger  combined  carbon  content.  On  account  of  its  hard- 
ness, this  steel  can  not  be  easily  machined,  but  -■■■ust  be  forged  to 
the  desired  shape.  Its  great  use  is  for  cutting  tools  --  roiwMrg 
tools . 


Chromium  steel  is  formed  by  adding  to  carbon  steel,  from 
1 en$  to  2.0$  chromium.  The  steel  thus  produced  is  extremely  hard 
and  also  self  or  air  hardening.  It  is  homogeneous  and  very  fine 
grained.  Because  of  the  rapid  oxidation  of  the  chromium,  the  steel 
deteriorates  rapidly  when  redressed.  'This  stool  is  used,  for  armor 
plate,  shells  and  metal  cutting  tools.  M The  boat  high  speed  tool 
steels  now  contain  from  5.0 $ to  6.0$  chromium." 

Manganese  steel  is  formed  by  adding  f erro-manganoso  to 
iron  or  steel.  The  proportion  of  carbon  present  is  high,  on  ac- 
count of  the  process  of  manufacturing.  The  influence  of  the 

manganese  is  not  always  the  sane,  thus  with  the  manganese 

192 


' 


■ 

■ 


. 


varying  from  a mere  trace  to  6*0$  and  with  carbon  not  exceeding 
0.4$  the  strength  and  ductility  of  the  compound  diminished  while 
the  hardness  and  brittleness  increase.  From  8$  to  10$  manganese 
the  strength  and  ductility  increase  rapidly.  With  12$  manganese, 
the  steel  becomes  we  air  but  on  increasing  the  manganese  to  about 
14.0 $ the  strength  of  the  metal  becomes  a maximum.  Above  14.0$ 
manganese  the  strength  of  the  steel  remains  fairly  constant  until 
about  17*0$  manganese  when  the  strength  diminishes  rapidly. 

Manganese  steel  is  in  general  free  from  blow  holes,  but 
is  difficult  to  cast  on  account  of  its  high  shrinkage  which  is 
about  two  and  one-half  times  as  great  as  cast  iron.  The  thermal 
conductivity  of  this  steel  is  low,  and  its  electric  resistance 
is  practically  constant  with  '.varying  temperatures.  It  possesses 
great  hardness  which  is  not  diminished  by  annealing,  and  also  a 
high  tensile  strength,  combined  with  great  toughness  and  ductility 
These  qualities  would  make  this  steel  the  ideal  metal  for  machine 
construction,  were  it  not  for  the  fact  that  its  great  hardness  pro 
vents  it  from  being  machined  in  any  way  but  by  abrasive  processes. 
It  in  the  most  durable  metal  known  to  resist  wear,  and  therefore 
is  used  extensively  for  steam  shovel  teeth,  dredge  pins,  plow 
points,  frogs,  switches,  crossings,  crusMng*rolls  for  ore,  rook 
screens,  gear  sprockets,  or  any  machine  mart  which  must  resist  a 
grinding  wear  in  dust.  The  best  composition  in  Mn  1^-15$:  0 not 
over  0.5$:  3 not  over  0.4$.  The  castings  should  always  be  anneal- 
ed* 

Vanadium  steel  is  formed  by  adding  a small  amount  of  vanadium 

192 


. 

. 


' 


13 


usually  loss  than  0.5$ — to  carbon  stsel.  Other  elements  ouch 
as  chromium  and  manganese  may  also  be  present.  The  vi' radium;  tends 
to  produce  brittleness;  this  steel  should,  therefore,  be  annealed, 
The  principal  use  of  vanadium  steel  is  for  metal  cutting  tools. 

Aluminum  steel  contains  only  a small  per  cent  of  that 
element,  since  it  easily  united  with  the  oxygen  and  passes  off 
with  the  slag  in  the  process  of  manufacturing.  Aluminum  increases 
the  strength  of  the  steel,  but  decreases  its  ductility.  This  steel 
is  not  in  extensive  use,  since  the  same  grade  of  steel  may  be  ob- 
tained by  employing  less  expensive  elements,  as  for  instance, 
silicon  and  manganese. 

Silicon  steel,  as  now  made,  contains  about  5.0$  Silicon 
and  shows  a very  fine  grained  structure.  It  nay  bo  made  very  hard 
hut  it  is  more  liable  to  crack  in  hardening  than  ordinary  steel. 
Silicon  steel  is  more  expensive  and  possesses  less  strength  than 

carbon  steel. 

9.  Cooper . Copper  in  a pure  state  is  a soft  motal  and 
is  extremely  ductile.  Good  castings  can  not  be  made  from  the 
pure  metal  because  of  blow  holes  and  great  shrinkage  in  cooling. 
Copoer  is  used  extensively  with  other  metals  to  form  alloys.  Other 
uses  are  for  certain  parts  of  electrical  machinery,  wire,  tubing, 
expansion  joints,  etc. 

10.  Alloys . Alloys  may  be  made  of  two  or  more  metals 

that  have  an  affinity  for  each  other.  The  compound  or  alloy  thus 

192 


. 

' 


' 


» f |K>  !;r'*l 


* 


' 


<■ 


14 


produced  has  properties  and  characteristics  which  none  of  the 
metals  possess.  The  principal  alloy  use:,  in  machine  construction 
may  bo  obtained  by  combining  two  or  more  oC  the  foil owing  mot.  Is; 
copper,  zinc,  tin,  lead,  antimony,  bismuth  and  aluminum. 

Brass  is  an  alloy  of  copper  and  zinc.  Its  composition 
may  vary  from  06  parts  cooper  and  24  parts  zinc  to  70  parts  co^mr 
and  30  parts  zinc.  Lead  is  sometimes  added*  especially  in  f-e 
cheaper  grades  of  brasses.  Brass  is  easily  worked  and  may  be 
made  to  take  a fine  finish.  For  this  reason  it  finds  extensive 
use  in  machine  parts,  especially  for  those  parts  not  requiring 
great  strength. 

Bronze  or  gun  metal  is  an  alloy  of  copper  and  tin.  Zinc 
is  some tines  added  to  cheapen  the  alloy  or  to  change  its  color  and 
to  increase  its  malleability.  Soft  bronze  contains  about  90  parts 
copper  and  10  parts  'tin,  while  hard  bronze  contains  02  parts  cop- 
per and  18  parts  tin.  Castings  stronger  than  those  made  from 
pig  iron  may  be  made  from  bronze.  The  softer  grades  are  m'  1 for 
cocks  and  fittings,  and  the  harder  grades  for  bearings  r ' bushings 
The  strength  and  toughness  of  bronze  are  greatly  increased  b^ 
rapid  cooling. 

Phosphor  bronze  varies  somewhat  in  composition*  but  in 
general  is  abCut  as  follows:  copper  80  parts,  tin  10  parts,  lead 
9 parts  and  phosphorus  1 part.  It  is  easily  cast  and  is  stronger 
than  cast  iron  castings.  Phosphor  bronze  is  used  for  bushings, 
bearings,  gear  wheels,  and  nay  be  drawn  into  wire  for  the  -manu- 
facture of  springs. 

Manganese  bronze  has  about  the  following  composition: 

192 


■ 


• 

' 


. 

■ 

• 

' 


■ 


* 


15 


copper  88  parts,  tin  8 parts,  and  manganese  d parts.  Several  qual- 
ities are  made  by  varying  the  percent  of  these  metals.  It  is 
stronger  than  phosphor  bronze  and  for  t;  is  reason  is  much  usod  for 
propel  lor  blade  castings.  It  is  also  usO''1  for  boa, rings  and  bush- 
ings, and  for  mining  screens  on  account  of  its  non-corrosive 
qualities . 

Babbitt  mot  a 1 L s om  etim  e s calls  d v h i t e o t a 1 or  rrh  j t o 
bra •••s,  is  an  alloy  of  copper,  tin  and  antimony  in  varying  pro- 
portions . Usually  these  proportions  are  as  follows  * tin  CA  to 
80  parts,  cooper  8 to  3 parts,  and  antimony  8 to  7 parts.  It  gives 
rise  to  less  friction  than  either  bronze  or  brass,  but  crushes 
more  easily.  It  is  not,  therefore,  suitable  for  use  in  severe  and 

heavy  work.  Its  only  use  is  for  hearings. 

1VS 


. 


■ 


I 


3YSICAL  CONSTANT; 


CO' 

I 1 


CQ 


p 

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c c c c c c 

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ft  CO  -H 

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t~i  eo 

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i 


AVERAGE  PHYSICAL  CONSTANTS , 


1C 


Ultima  te 
t ensile 
strength 

Ultimate 

compressive 

strength 

Ultimate 

shearing 

strength 

! 

Elastic  {Ultimate 

limit.  {Flexural 

j Strength 

Modulus  or 
coef.  of 
Eleotricity 

Shearing 
Modulus  of 
Electricity 

Height 

LI . per 
sq . in. 

Lb . per 
sq.  in. 

Lb  « per 
sq . in . 

T 

Lb . per  1 

sq.  in. 

Lb . per 
sq.  in. 

Lb . per 
sq.  in. 

Lb . per 
oq.  in. 

Lb . per 
cu . ft . 

Hard  Stool 

100,000 

180, ono 

ro , ooo 

50,000 

110,000 

50,000,000 

18,000,000 

•190 

Structural  Steel 

50,000 

00,000 

50 f nor 

c 5,o no 

50,000,000 

18,000,000 

490 

V r ought  iron 

50,000 

50,000 

/ n ooo 

2 5 , ooo 

85,000 ,000 

10,000,000 

-IPO 

5 5,000  Ton. 

Cast  iron 

P.0,000 

90.000 

20,ooo 

)£0 ,onr  Oom. 

35,000 

15,000,000 

0 ,.000,000 

4 50 

Copper 

30 , 000 

15  r0 00  ,.000 

6 ,-000,-900 

55  0 

Timber  Y'ith  Grain 

10,090 

0,000 

coo 

3 ,0°0 

9,000 

1 r50 0.,  090 

40 

Across  Grain 

3,000 

400,000 

•V) 

Concrete 

300 

5 , 000 

1,000 

1,000 

70Q 

5 ,-000  r000 

150 

Stone 

6 , 000 

1,500 

2 ,nno 

P ,000 

6,000,000 

ICO 

Brick 

. 

3,000 

. - 

1,000 

1,000 

BOO 

8,000,000 

185 

SAFETY  FACTORS. 


Material  For  steady  Stress  For  Varying  Stress  For  Shocks 
(Buildings)  ^ Bridges)  (Machines) 


Hard  Steel 

5 

6 

15 

Structural  Steel 

4 

A 

C 

10 

Wrought  Iron 

4 

6 

10 

Ca3t  Iron 

c 

10 

20 

Timber 

6 

TO 

’fj 

CHAPTER  II 


17 


Strength  of  Mater ials . 

The  object  of  a machine  is  to  transmit  motion  through 
its  various  links,  to  some  particular  part  where  useful  vr ork  is 
to  be  done*  The  transmitting  of  this  motion  gives  rise  to  forces 
which  must,  be  resisted  by  the  parts  of  the  machine  through  which 
the  force  is  acting.  In  order  to  safely  withstand  these  forces, 
each  machine  part  must  be  constructed  in  accordance  with  certain 
laws . These  laws  may  bo  the  result  of  theoretical  investigation 
or  they  may  be  obtained  from  existing  conditions  of  design.  For- 
mulas for  the  size  of  the  machine  parts  obtained  from  this  latter 
source  are  called  empirical  - In  the  design  of  machine  parts 
many  of  the  formulas  used  are  more  or  less  empirical.  This  is 
especially  true  of  the  greater  number  of  the  formulas  usod  in 
this  elementary  design  in  machine  design.  The  student  is  not 
far  enough  advanced  in  his  applied  mathematics  to  enable  him 
to  derive  but  few  of  the  formulas.  He  should,  however,  familiar- 
ize himself  with  the  meaning  of  the  various  symbols  occurring 
in  the  formulas,  and  should  also  bo  able  to  use  readily  any 
formula  occurring  in  these  notes. 

1 1 . External  For c es,  Stress  ^ a rd  d trains.  - C on s i de r 

an  iron  rod  whose  cross  section  is  A square  inches,  su on ended 

from  so^'6  fixed  point,  and  sup  nr  o ting  at  its  lower  end  a load 

or  weight  of  P pounds.  Evidently  the  pull  downwards  at- the  end 

of  the  rod  is  P pounds,  and  if  equilibrium  is  to  be  maintained , 

the  rod,  in  any  section  near  the  bottom,  must  exert  a pull  eoual 

102 


i 


' 

' 


■ 


■ (Hi. 


1G 

to  P and  in  the  opposite  direction.  This  force,  which  arises 
in  the  rod  due  to  the  pull  of  the  weight  P,  is  called  a stress, 
and  map  be  defined  as  the  internal  resistance  which  the  mole- 
cules of  the  rod  offer  to  the  force  P.  In  other  words , the 
internal  resistance  offered  by  a body  to  any  force  tending  to 
overcome  the  force  of  cohesion  is  called  a stress.  In  the  above 
rod  if  S represents  the  stress  per  square  inch  of  cross  section, 
then  evidently; 

Area  of  cross  section  of  rod  multiplied  by  unit  stress 
= total  stress  induced  in  section,  but  the  total  stress  induced 
in  section  is  due  to  the  pull  of  the  external  force,  hence 

AS  = P (1) 

From  (l)  if  we  know  the  magnitude  of  the  external 
force  and  the  value  of  the  stress  per  square  inch  we  nay  easily 
find  the  area  of  the  section. 

Stresses  are  usually  measured  in  pounds  or  tons  per 
unit  area.  In  this  course  the  unit  of  area  will  be  the  square 

inch  and  the  unit  stress  or  intensity  of  stress  will  be  given 
in  pounds  per  square  inch.  The  values  of  the  unit  stresses, 
are  obtained  by  experiment,  that  is,  by  testing  pieces  to 
destruction  in  testing  machines.  For  values  of  these  stresses 
for  various  materials  see  Table  I. 

Strain,  - A body  subjected  to  an  external  force  no 
matter  how  small,  undergoes  a deformation  or  change  of  form, 

tiie  amount  of  which  is  called  a strain . Thus  a rod  10  inches 

\ 

long  has  suspended  from- it  a weight  such  that  its  length 

while  supporting  the  ’weight  is  10.01  inches. 

192 


The  total 


* 


. 


. 


- 


strain  is  therefore  *01  inches  and  the  unit  strain  or  strain 


per  unit  of  length  is  .001.  Within  certain  limits  strains 
are  directly  proportional  to  the  stresses  which  produce  them. 

Let  1 = length  of  body 

e = total  deformation  or  elongation  of  body 
s = unit  of  strain 

then  e = Is . ' (2) 

There  are  three  kinds  of  simple  stresses  induced  in 
a machine  part  by  the  external  forces  acting  upon  the  part. 

They  are  as  follows: 

(1)  Tensile,  in  which  the  externa]- force s are  tec  ling 
to  pull  the  body  apart.  As  an  illustration  of  this  stress  con- 
sider the  rod  mentioned  above.  The  rod  may  fail  by  being  nulled 
apart.  Here  the  external  forces  are  acting  away  from  each  oth'.T. 

(2)  Compressive , in  which  the  external  forces  are  ten- 
tending to  crowd  the  parts  of  the  body  together.  The  piston  rod 
of  an  engine  on  the  forward  stroke,  legs  for  the  support  of 
lathes,  columns,  are  examples  of  machine  parts  subjected  to 
compressive  stress. 

(3)  Shearing,  in  which  the  external  forces  are  act- 
ing on  the  body  in  parallel  lines  very  near  each  other,  in  the 
opposite  sense.  A good  illustration  is  the  punching  of  a hole  in 
a plate,  or  the  shearing  of  bars  or  plate  : in  a shearing  machine. 

In  addition  to  the  simple  stres -es  given,  above,  may 
also  be  mentioned  bearing  stress.  This  is  a form,  of  compres- 
sive stress,  and  is  caused  by  two  surfaces  pressing  or  bearing 

192 


■ 


. 


. 


- 


2.0 

against  each  other « Plato  edges  on  rivets  or  nine,  cotter  edges 
and  keys  in  key  ways  are  all  illustrations  of  machine  warts  in 
which  bearing  stress  is  induced. 

There  are  two  kinds  of  stresses,  viz:  torsion  and  bond- 
ing. A brief  discussion  of  these  will  be  given  later. 

Experiments  have  shown  that  when  a body  in  subjected 
to  a small  stress  a small  strain  is  produced,  and,  upon  removal 
of  the  external  force  causing  the  stress,  the  body  returns  to 
its  original  shape.  This  property,  belonging  to  the  body,  i3 
called  elasticity . All  bodies  are  more  or  less  elastic,  and 
within  certain  limits  the  strain  is  directly  proportional  to 
the  stress.  That  is,  if  a force  produces  a stress  of  10000 
pounds  and  a corresponding  strain  of  . O'*!  inches,  thin  doubling 
the  force  would  produce  twice  the  stress  wit!  twice  the  strain, 
or  taking  one-half  the  external  force,  the  corresponding  stress 
would  he  5000  pounds  and  the  strain  <-0015  inch.  The  external 
force  acting  upon  the  body  may,  however,  bo  of  such  a magnitude 
that  the  resulting  stress  produces  a strain  which  is  part ly  per- 
manent. Under  these  conditions  the  strain  is  called  a set . 

The  strain  is  no  longer  proportional,  to  the  stress,  which  produced 
it.  If  tho  external  force  be  increased  then  the  stress  will  be 
increased  and  the  strain  will  rapidly  increase  until  the  body  is 
broken.  The  stress  at  the  point  where  the  strain  becomes  perma- 
nent is  called,  the  elastic  limit.  Evidently  no  machine  part 
should  be  loaded  so  a-:  to  produce  a stress  near  the  elstic  limit. 

The  coefficient  of  elasticity  of  a body  is  the  ratio 

192 


. 


I ' ! 


21 


of  the  unit  stress  to  the  unit  strain. 

Let  S = the  unit  stress 
s = the  unit  strain 
E = the  coefficient  of  elasticity. 

Then  by  the  definition  and  combination  of  (l)  and  (2)  we 


have 


E 


(3) 


If  in  (3)  E be  regarded  as  a force,  also  that  a = 1 
and  A » unit  area,  we  should  have  E = P.  The  interpretation 
of  this  is  left  as  an  exercise  for  the  student. 


The  ultimate  strength  of  a body  is  that  unit  stress 
which  is  just  sufficient  to  break  it. 

12.  Stresses  due  _t o_  Bonding, .-If  w e c on s i d e r t h e 
forces  acting  in  any  machine  we  shall  find  thi  t some  of  them 
act  without  leverage,  thus  producing  direct  stress  in  the  machine 
part.  As  an  example,  consider  the  piston  rod  of  an  engine.  The 
steam  pressure  is  the  external  force  transmitted  through  the  rod, 
thus  producing  a stress  within  the  rod,  the  magnitude  of  which 
is  equal  to  the  external  force.  The  stress  in  uniformly  distri- 
buted over  the  cross  section  since  the  resultant  of  the  steam 
pressure  acts  through  the  geometric  axis  of  the  rod.  This  stress 
is  called  a direct  stress. 

Again,  if  wo  continue  our  analysis  we  shall  find  that 

some  of  the  forces  act  with  a leverage,  thus  producing  bending 

in  the  machine  part.  As  an  example,  consider  the  crank  of  the 

steam  engine.  The  force  transmitted  through  the  connecting  rod 

to  the  crank  through  the  crank  pin  produced  a 

192 


moment  about  the 


' 


op 

center  of  the  crank  shaft®  The  stresses  thus  induced  in  the 
shaft  are  not  direct  stresses,  and  the  determination  of  "d>ich  - 
require  some  knowledge  of  the  theory  of  beams. 

15.  Beams,  - Consider  the  bar  in  Pig.  1 resting  on  the 
support  R^  and  Rg  and  supporting  a load  W at  a distance  x from 
the  left  support.  Without  the  load  W the  bar  will  be  in  the 
position  indicated  by  the  full  liners,  but  supporting  the  load 
the  bar  will  tend  to  assume  the  p o s i t i on  indicated  by  the  dotted 
lines.  If  we  assume 9 as  we  may,  that  the  bar  is  made  up  of  an 
infinite  number  of  fibers  running  the  long  way  of  the  bar,  then 
those  fibers  lying  in  the  top  surface  containing  the  line  ran, 
will  be  shortened  or  compressed,  and  those  fibers  in  the  surface 
containing  the  line  op  will  be  lengthened  or  stretched.  In  other 
words,  the  upper  fibers  will  be  subjected  to  a compressive  stress 
and  the  lower  fibers  to  a tensile  stress.  Evidently  somowK-vro 
between  the  upper  and  lower  fibers  is  a surface  the  fibers  in  which 
are  neither  shortened  or  lengthened.  Such  a surface  is  called 
the  neutral  surface  and  always  contains  the  center  of  gravity, 
provided  the  loading  is  as  shown.  Referring  to  Pig.  1,  the  lino 
qr  represents  the  neutral  surface. 

Let  us  consider  the  forces  acting  on  the  bar  in  Pig. 

1.  The  weight  W acting  downward  causes  the  supports  at  R-j 
and  R0  to  exert  an  upward  pressure  on  the  bar.  To  find  the 

r-Z> 

value  of  these  upward  pressures  or  reactions,  we  employ  the 

principle  that  the  sum  of  the  moments  of  r.  11  the  forces  witj 

respect  to  any  point  is  equal  to  zero,  nrovided  the  forces  are 

in  equilibrium.  Since  this  is  true,  with  resnect  to  any  noint, 

19? 


. . 


. 


■ 


• • 


, 


■ 


. 


let  up.  take  moments  , about 
action  of  either  or  R0 . 
right  reaction  Rr> , then  we 
of  the  bar 


a point  containing  tho  lino  of 
Suppose  we  take  moments  about  the 
shall  have,  if  1 equals  the  length. 


Ri1 


or 


W(l-x)  = 0 
W(l-x) 

1 


also  if  we  take  moments  about  the  left  support,  we  shall  have 


Here  we  have  considered  forces  acting  upward  as  positive,  and 
those  acting  downward  as  negative. 

Let  us  next  determine  the  bending  moment  in  the  bar 
due  to  the  load  W at  any  section  as  ab.  The  bonding  moron t is 
measured  by  the  resultant  moment  of  the  external  force  on  either 
side  of  the  section  ah . We  shall  consider  tho  forces  to  the 
left  of  the  section  ah.  The  only  force  on  the  left  of  the  sec- 


tion ab_  is  the  reaction  and  tho  tendency  of  this  force  to 

cause  dending  in  tho  section  depends  upon  its  magnitude  and  its 

distance  from  the  section.  If  wo  consider  the  section  cd,  then 

we  shall  have  two  forces  to  the  left  of  the  section  - tho 

reaction  R]_  and  tho  weight  W,  The  tendency  of  1?^  is  to  cause 

rotation  of  the  bar  in  a clockwise  direction  about  a pdint  in 

the  section  cd  and  the  magnitude  of  this  tendency  to  rotate  the 

bar  is  measured  by  the  moment  of  tho  force  R-j  with  respect  to 

the  point  in  the  section.  The  tendency  of  W is  to  cause  rotation 

of  the  bar  in  a counter  clockwise  direction, 

192 


and  the  magnitude 


. 


- 


nA 

of  this  tendency  to  rotate  is  also  measured  by  the  moment  of 
W with  respect  to  the  point  in  the  section.  The  algebraic  sum. 
of  these  tendencies  to  produce  rotation  about  a point  in  the 
section  od  is  called  the  bending  moment , or  the  algebraic  sum 
of  the  moments  of  the  forces  on  the  left  of  the  section  with 
respect  to  a point  in  that  section  called  the  bending  moment . 
Expressing  this  bending  moment  by  means  of  a formula,  we  shall 
have 

Bending  moment  = &ixi  ~ Wxg  (6) 

14 o Bee j sting  moment,  - Let  us  consider  that  part  of 

the  bar  to  the  right  of  the  section  cd.  to  be  removed  as  shown 

in  Fig.  2 < The  fibers  in  the  upper  part  of  the  bar  an  wo  have 

seen  are  in  compression  while  those  fibers  in  the  lower  part 

are  in  tension.  Evidently  the  compressive  stresses  tend  to 

produce  a rotation  of  the  upper  part  of  the  bar  in  a counter 

clockwise  direction,  while  the  tensile  stresses  tend  also  to 

produce  a rotation  of  the  lower  part  of  the  bar  in  a counter 

clockwise  direction.  The  tension  below  nd  the  compression 

above  have  resultants  R and  R whose  moment  is  Em,  The  moment 

of  one  of  these  forceps  with  respect  to  the  line  of  action  of  the 

other  is  known  as  the  moment  of  resistance  of  the  section. 

Evidently,  for  equilibrium,  the  bonding  moment  must  equal  the 

resisting  moment,  or 

R^xq  - Wxp  = Em  ( 7 } 

15.  - A bar  subject  to  the  loading  shown  in  Fig.  1 is 

called  a simple  beam.  A cantilever  beam  is  one  having  one  end 

fixed  and  the  other  free.  A restrained  beam  is  one  which  has 

both  ends  fixed.  Each  of  these  beams  with  the  loading  is 

192 


. 


' 


r . I 


represented  in  Fig,  3.  This  figure  shows  the  beaT”is  as  all  sup- 
porting loads  concentrated  at  a point.  Instead  of  this  arrange*- 
ment , the  beam  may  have  the  loads  distributed  over  its  length, 
either  uniformly  or  varying  according  to  some  given  law,  or 
several  concentrated  loads c 

Exercise:  Visit  the  shops  and  laboratories  and  write 

down  the  name  of  at  least  one  machine  part  loaded  as  a beam; 


that  is, 

(a)  One  piece  acting  as  a simple  beam  with  a concentrated 

load . 


ti  ti 


" load  uniforml 


cantilever  bean  with  a concen- 


f!  II 


restrained 


(b)  " " " 

distributed . 

( c ) One  " 11 

trated  load. 

( d ) One  " " 

formly  distributed. 

(e)  One  piece  " 
trated  load. 

( f } One  " *' 

formly  distributed. 

The  student  in  his  shop  work  should  observe  the  charac 
ter  of  the  loading  of  the  parts  of  the  machines  with  which  he 
works.  The  beam  is  of  frequent  occurrence  in  machine  construc- 
tion. and  one  should  be  able  at  once  to  know  the  character  of 
the  loads  in  order  that  the  bending  moments  may  bo  accurately 

determined  and  the  part  thus  properly  proportioned. 

192 


if  il 


II  If 


If  II 


It  II 


load  uni 


concen- 


" load  uni- 


. 

. 


; 

' 


. 


. 


' | 


The  bending  moment;  M may  bo  calculated  from  formulas 
given  in  table  in  Kent,  p.  268.  In  the  first  column  is  given 
the  character  of  the  beam  and  the  nature  of  the  load.  The 
column  headed  "Maximum  moment  of  Stress"  gives  the  value  of 
the  bending  moment  M while  the  column  headed  "Moment  of  Rupture 
gives  the  resisting  moment,. 

16 . - We  have  seen  that  for  equilibrium  the  bending 
moment  must  equal  the  resisting  moment  (Lq.  7),  Let  us  next- 
find  an  expression  for  its  value  in  terms  of  the  dimension  of 
the  section. 

Let  S = tensile  or  compressive  fiber  stress  upon  fiber 
farthest  away  from  the  gravity  axis, 

c = distance  from  gravity  axis  to  most  remote  fiber,  in 
inches . 

5 = distance  from  gravity  axis  to  any  fiber  whose  area 

equals  a sq.  in. 

Then  by  the  following  law,  sometimes  called  Hookes  Law 
which  is  "the  amount  of  elongation  or  compression  in  any  fiber 
is  directly  proportional  to  its  distance  from  the  gravity  axis" , 
we  have 

§.  = stress  per  sq.  in.  at  a distance  of  1"  from  gravity  axis, 
c 


c _ 
O s 

C 


It  It  it  if  ii  it 


ll 


if  :j  ii  it 


It  I! 


o 2 

(—  Ba)r  = a = moment  of  this  stress 

r*  ' 

v-'  ^ 

is  "a"  about  the  gravity  axis. 


in  the  fiber  who s ■_ 


area 


How  we  consider  the  piece  as  being  made  up  of  an 

infinite  number  of  fibers  whose  area  is  "a" : hence  the  sum  of 

192 


■ 

' 


. .. 


- 


' 


■ 


. 


°7 


all  the  moments  of  the  stresses  induced  in.  each  fiber  would  be 
the  sum  of  all  the  expressions: 


ba 


or 


Sa  3s 


> a5 since  S and  c are  con- 


C \ 0 c ^ 

p 

stants.  But  the  expression > a Z0  being  the  sum  of  the  products 

formed  by  multiplying  each  of  the  elementary  strips  by  the  square 

of  its  distance  from  the  gravity  axis  is  called  the  moment  of 

inertia  of  that  section  with  t.respect  to  the  gravity  axis.  The 

moment  of  inertia  in  usually  denoted  by  I.  We  have  therefore 

q p SI 

resisting  moment  Em  = _ > aZ  = — 7 or  denoting  the  bending 

c 0 

moment  by  M we  have 


M = 


31 


(8) 


In  (8)  the  ratio  =*— • is  called  the  section  modulus, 

c 

values  of  which,  for  various  sections  may  be  found  from  tables 

in  Kent,  p.  249. 


192. 


■ 


..  . . 


■ 


_ 


• • 


CHAPTER  III 


28 


FASTENINGS 0 
Riveted  Joi n ts , 

17.  Rivets.  - The  most  common  means  of  uniting  plates 
as  used  in  boilers,  tanks  and  structural  work  is  by  means  of  ri- 
vets. A rivet  is  a round  bar  consisting  of  an  upset  end  called, 
the  head  and  a long  part  called  the  shank.  It  is  a permanent 
fastening,  removable  only  by  chipping  off  the  head.  Rivots 
should  in  general  be  placed  at  right  angles  to  the  forces  which 
cause  them  to  fail.  The  greatest  stress  thus  induced  in  then 
is  that  of  shearing.  If  rivets  are  to  resist  a tensile  stress 
a greater  number  should  be  used  than  when  they  are  resisting 
a shearing  stress. 

Rivets  are  made  of  wrought  iron  and  soft  steel  and  are 

formed  in  suitable  dies  while  hot  from  round  bars  out  to  length. 

The  shank  usually  has  parallel  sides  for  about  one-half  its 

length,  the  remaining  length  tapering  every  slightly.  When  used 

the  rivets  are  brought  up  to  a red  heat,  placed  in  the  holes 

of  the  plates  to  be  connected  and  a second  head  formed,  oith/  r 

by  hand  or  machine  work.  In  the  former  hammers  are  used;  tf  e 

head  of  the  rivet  being  held  firmly  against  the  under  side  of 

the  plate.  In  the  latter,  the  rivet  is  pressed  between  two 

dies.  Generally  speaking,  machine  riveting  is  better  than  hand 

work,  as  the  hole  in  the  plates  in  nearly  always  filled  with  the 

rivet  body,  while  in  hand  work  the  effect  of  the  blow  does  not 

appear  to  reach  the  interior  of  the  rivet,  thus  producing  n) 

192 


. 


movement  of  the  metal  in  the  rivet  hole. 


18-  Rivet  Holes.  For  the  sake  of  economy  rivet  hoi -s 
are  usually  punched-  There  are  two  serious  objections  to  thus 

forming  the  hole.  The  metal  around  the  holes  is  injured  by  the 
lateral  flow  of  the  metal  under  the  punch.  This  may,  however, 
be  remedied  by  punching  smaller  holes  and  then  reaming  them  to 
size.-  Next,  the  sjmcing  of  the  holes  in  the  two  parts  is  not 
accurately  done  in  the  case  of  punching  so  that  it  becomes  neces- 
sary to  either  ream  out  the  holes  (in  which  case  the  rivets  may 
not  completely  fill  the  hole  thus  enlarged)  or  use  a drift  pin. 
The  drift  pin  should  be  used  only  with  light  weight  harpers  * 

The  size  of  the  rivet  hole  is  about  l/l6  inch  larger 
than  the  rivet.  This  is  subject,  however,  to  some  variation, 
depending  upon  the  class  and  character  of  the  work.  This 
clearance  space  allows  for  some  inaccuracy  in  punching  the 
plate  and  also  permits  clearance  space  for  driving  the  rivet 
when  hot » 

Drilling  the  holes  is  the  best  method  of  perforation 
of  plates.  The  late  improvements  in  drilling  machinery  has 
made  it  possible  to  accomplish  this  work  with  almost  the  same 
economy  as  in  punching.  The  metal  is  not  injured  by  the  drillin 
of  holes;  indeed  there  are  tests  which  show  an  increase  in  unit- 
strength  of  the  metal  between  the  rivet  holes . 

19-  Forms  of  Rivets.  - Rivets  are  made  from  a very 

tough  and  ductile  quality  of  iron  and  steel,  They  are  formed 

in  dies  from  the  round  bars  while  hot  and  in  this 

192 


c ond  3 1 i on 


. 


. 


4 


■ 


- ■ 


■ 


. 

. 

' . 

. 

■ ' 

• 

■ . . 

r50 


are  called  rivet  blanks,,  The  rivet  blank  is  composed  of  two 


parts,  viz.,  the  head  and  the  shank*  For  convenience  the  head, 
which  is  formed  during  the  process  of  driving  is  called  the 
point.  The  amount  of  shank  necessary  to  form  the  point  depends 
upon  the  diameter  of  the  rivet . Since  the  length  of  rivet  is 

measured  under  the  head,  the  length  required  is  equal  to  the 
length  of  shank  necessary  to  form  the  point  plus  the  thickness 
of  the  plates.  The  thickness  of  the  plates,  or  the  distances 
between  the  head  and  point  after  the  rivet  is  driven  is  called 
the  grip  of  the  rivet . To  find  the  length  of  rivet  required  for 
uniting  plates,  add  to  the  combined  thickness  of  plates  a 

length  equal  to  1 1/2&  for  steeple  points  (see  Fig.  4 
button  points  (Fig.  4,  b)  add  to  combined  thic^nsr 


A - ) 

pj1  r\  71-* 

c r Cl 

of  pi 

at  os  a 

. .p  a 

{ '"'t  C\  Uj 

Fig.  4,  c).  Lengths  of  rivets  should  always  he  taken  in  quarter 
inch  lengths  on  account-  of  stock  sizes.  Any  length  up  to  five 
or  six  inches,  however,  may  be  obtained,  but  the  odd  sizes  will 
cost  more  than  the  standard  sizes. 

TO.  Forms  of  Heads.  - Rivets  with  many  different  forms 
of  heads  may  be  found  in  mechanical  work,  but  the  ones  in  general 
use  in  boiler  work  are  only  three  viz . , cone  head,  button  head, 
and  countersunk  head.  These  are  shown  in  Fig.  4,  (a),  (b),  and 
(c)  respectively.  The  proportions  advocated  by  different  manu- 
facturers vary  somewhat.  The  proportions  given  in  Fig.  4 are 
those  used  by  the  Champion  Rivet  Company.  The  steeplo  poirr  , 

Fig.  4 (a)  is  one  easily  made  by  hand  driving  and  is,  therefore , 

rauoh  used.  This  form,  however,  on  account  of  the  thinness  of  the 

192 


' 


' 

■ 


1 


pn 

edges,  is  weak  to  resist  tension  and  should  not  therefore  be 
used  on  important  work. 

The  cone  head,  Fig.  4 (&),  is  one  of  tgreat  strength 
and  is  used  a great  deal  in  boiler  work.  It  is  not  generally 
used  aa  a form  for  the  point  on  account  of  difficulty  in  driving. 
The  button  head  type,  Fig.  4 (b)  is  widely  used  for  points  and 
may  be  easily  formed  in  hand  work  by  the  aid  of  a snap-,.  A snap  is 
much  used  in  forming  points.  It  is  a piece  of  steel  with  a 
forming  die  in  one  end.  The  die  is  placed  over  the  rivet  and  the 
snap  struck  with  a heavy  hammer,  in  this  manner  an  almost  perfect 
point  may  be  formed. 

The  countersunk  point  weakens  the  plate  so  much  that 
it  is  used  only  when,  projecting  heads  would  be  objectionable, 
as  under  flanges  of  fittings,  in  the  direct  line  of  the  play  of 
the  flames.  Its  use  is  sometimes  imperative  for  both  heads  and 
points,  but  it  should  be  avoided  whenever  possible.  The  counter- 
sink in  the  plate  should  never  exceed  3/4  of  the  thickness  of 
the  plate,  and  for  that  reason  the  height  of  the  rivet  point 
is  generally  from  l/l6  to  l/8  of  an  inch  greater  than  the  depth 
of  the  countersink.  The  point  then  projects  by  that  amount 
or  if  the  plate  is  required  to  be  perfectly  smooth,  the  point 
chipped  off  level  with  the  surface. 

SI.  Riveted  Joints.  - There  are  two  methods  of  con- 
necting bars  abd  plates  by  means  of  rivets.  The  arrangement 
by  which  the  edge  of  one  plate  overlaps  the  edges  of  the  o\  her 
is  called  the  lap  joint.  Such  a joint  may  ho  riveted  up  as 

shown  in  Fig.  5 with  only  one  row  of  rivets,  in  which 

192 


0O  r*  ,-TI 

a,  o 


. 


15;  " ' jj , . 


■ 


■ 


cal led 


single  riveted  lap  joint  „ With 


the  arrangement  is 


two  rows  of  rivets  the  joint  is  .known  as  a double  riveted  lap 
joint , and  with  three  rows  a triple  riveted  lap  joint.  When 
more  than  one  row  of  rivfets  is  used  they  may  be  arranged  as 
shown  in  Pig.  6,  (b)  or  (a).  In  the  former  the  joint  is  chain 
riveted  and  in  the  latter  zig-zag  rivetedc 

When  the  plates  or  bars  butt  against  each  other  and 
are  joined  by  overlapping  plates  or  straps  the  connection  is 
called  a butt  joint.  Such  a joint  may  have  one  plate  on  the 
outside  or  one  plate  on  the  outside  and  one  on  the  inside;  may  be 
single,  double  or  triple  riveted  and  may  have  chain  or  zig-zag 
riveting.  Fig.  7 shows  a single  riveted  butt  joint  with  two 
cover  plates  . 


22.  Failure  of  Riveted  Joints.  - Joints  may  fail  in 
one  of  several  ways,  as  shown  by  the  following  examples. 

(a)  Shearing  of  the  Rivet.  - In  the  case  of  all  lap 
joints  and  butt  joints  with  one  strap,  the  rivets  tend  to  fail 
along  one  section,  while  in  butt  joints  with  two  straps  failure 
tonds  to  take  place  along  two  sections.  Thus  in  Pig.  P (a) 
the  tendency  would  be  for  the  rivets  to  fail  along  the  section 
mm  and  after  failure  the  conditions  would  be  represented  by 

Fig.  S (b).  Such  a rivet  is  said  to  be  in  single  shear.  If  P is 
the  force  tending  to  pull  the  plates  past  each  other,  S„  the 

o 

ultimate  shearing  strength  of  a square  inch  of  section  of  the 

rivet  and  A the  cross  sectional  area  of  the  rivet,  then  from  (l) 

192 


. 


' 


• 

l 

p 


(9) 


'7ta2ss 

4 

Exercise , Sketch  a butt  joint  with  two  straps 
and  deduce  formula  for  shearing  resistance  of  one  rivet. 

(b)  Crushing  the  Plate  or  Rivet.  - If  the  rivet 
should  be  strong  enough  to  resist  the  shearing  force  then 
the  plate  may  fail  bj‘  crushing  and  wrinkling;  as  shown  at  A in 
Fig.  9.  The  resistance  to  crushing  offered  by  any  small  portion 

MIT , Fig.  10  on  the  circumference,  equals  the  projection  of  MI! 

on  the  diameter  perpendicular  to  the  line  of  action  of  the  force. 

The  total  resistance,  therefore,  would  be  the  sun  of  all  such 

projections  times  tho  thickness  of  the  plate.  Evidently  the  sum 

of  all  such  projections  would  be  equal  to  the  diameter  of  if  e 

/ 

rivet  and  the  total  resistance,  therefore,  would  be  dtSc  in  which 

d = diameter  of  the  river,  t ■ = thickness  of  plate,  and  S0  = the 

ultimate  crushing  strength  of  plate.  Since  this  must  equal  the 

force  P tending  to  cause  crushing,  we  have 

P = dtS  (10) 

c 

(c)  Bursting  of  Plate.  - A joint  may  fail  by  split- 
ting of  the  plate  opposite  each  rivet,  as  shown  at  B in  Fig, 

9.  This  manner  of  failure  may  be  prevented  by  having  a suf- 
ficient distance  from  the  rivet  to  the  edge  of  the  plate. 

It  has  been  found,  experimentally,  that  if  this  distance  is 
at  least  equal  to  the  diameter  of  the  rivet  then  failure  will 
not,  in  general,  take  place  in  this  way. 

(d)  Tearing  of  the  Plate.  - The  nlateo  may  bo  torn 

195 


, 


■ 


• . 


* 

. 


- 


or  pulled  apart  along  the  line  of  rivets  as  shown  at  CD,  Fig. 
9.  Evidently  the  least  resisting  area  offered  by  the  plate 
to  tearing  would  be  the  net  area  of  the  plate  along  the  lino 
of  rivets.  If  p = pitch  of  rivets  or  distance  from  center 
to  center  of  rivet  holes,  t = the  thickness  of  plate,  and  St 
= ultimate  tensile  strength  of  plate,  then  the  resistance  of 
the  plate  to  tearing  would  be  (p~d)tS+,  or  equating  to  exter- 
nal force, 

P - ( p-d ) tS't  (11) 

fe)  Shearing  of  the  Plate.  - A plate  night  fail 
by  shearing  along  the  linos  in  front  of  the  rivet,  as  shown 
at  E in  Fig.  9.  Failure  in  this  way  is  not  likely  to  occur. 

To  find  the  resistance  of  the  plate  to  shearing,  if  a equals 
the  margin  or  distance  from  center  of  rivet  hole  to  the  edge 
of  plate,  then  the  shearing  resistance  offered  by  the  plate 
would  be  2atS0 , and  the  force  required  to  cause  failure  would 
be 

P = 2at8  (12) 

8 

23*  Regarding  the  use  of  the  lap  joint  in  the  boiler 

construction,  experience  has  shown  that  such  joints  are  dan- 

gerous on  account  of  the  liability  of  hidden  cracks.  These 
defects  are  due  chiefly  to  the  fact  that  the  lines  of  action 
of  the  forces  in  the  two  plates  do  not  coincide.  A couple  is 

thun  set  up  which  causes  the  plate  to  herd,  and  nay  therefore 

cause  the  plate  to  crack.  Some  boiler  Inspectors  condemn  +ho 
use  of  lap  joints  in  boiler  construction. 

Theoretical  formulas  have  been  deduced  which,  give 

proportions  of  boiler  joints  for  uniform  strength.  These 

192 


- 


. 


' . ■ 


. 


’ 


■ 


35 


formulas,  however,  are  not  adhered  to  in  practice  for  economic 

reasons.  . The  usual  average  practice  for  joints  perhaps  in 

this  country  are  those  proportions  advocated  by  the  Hartford 

Steam  Boiler  Inspection  and  Insurance  Co.  These  designs  may 

be  found  in  the  bach  of  Scully's  Steel  and  Iron  Co's.  Stock  List. 

108 


• 1 


:,NMi  IMFf  v,v  Ifj  gjgjtv;?;  . Ippi ► / ■ i“ 


• • 


■ 


CHAPTER  IV 


56 


Fa  sjb  e Ti irigs  . 

Bolts  and  Huts « 

24.  Bolts,  - A bolt  is  a round  bar  on  which  lias  been 
formed  a helical  projection  or  thread.  Usually,  only  one  end 
of  the  rod  is  fitted  with  a thread  while  the  other  is  upset  to 
form  the  head.  When  used  as  a fastening  a hollow  cylindrical 
part  which  has  threads  formed  on  its  inner  surface  in  used.  This 
part  is  called  the  nut.  Sometimes  one  or  two  of  the  parts  to  ho 
united  may  have  the  threads  in  the  hole  through  which  the  bolt 
passes.  In  such  cases  the  nut  as  an  extra  part  is  not  required. 

The  object  of  bolts  in  to  fasten  machine  parts  firmly, 
and  also  allow  the  part3  to  be  easily  separated,  or  disconnected. 
The  bolt  passes  through  the  parts  to  be  connected  and  when  the 
nut  is  screwed  down,  surface  compression  is  caused  in  the  parts 
thus  united,  while  the  parts  themselves  react  on  the  head  and  nut, 
producing  tension  in  the  bolt.  Ordinarily  bolts  must  resist  a 
tensile  force,  although  they  may  in  some  cases  be  required  to 
resist  a shearing  f oroe . To  find  the  bolt  diameter  for  a given 
load  use  formula  ( 1 ) . In  this  case  should  be  taken  rather  low 
especially  in  bolts  less  than  one  inch  in  daimeter.  A equal's 
the  net  area  at  the  base  of  thread.  For  United  States  Standard 
Threads  the  diameter  of  bolt  for  a given  area  at  base  of  thread 
is  given  in  table  in  Kent,  p,  205.  For  other  threads  calculations 
must  be  made  for  the  diameter  of  bolt  when  area  at  the  base  of 
thread  is  known. 

192 


i 

. 


' 


' 

• 

37 


The  commercial  forme  in  which  bolts  aro  made  are 
as  follows: 

( a ) Thr ou g h B o 1 1 s , usually  rough  with  square  lioads 
and  nuts  (Pig.  11,  a)  or  hexagonal  heads  and  nuts  (Pig.  11,  b). 
The  standard  lengths  of  "through  bolts"  as  givsn  by  the  manu- 
facturers T catalogue  are  as  follows:  between  1"  and  5"  length's 
vary  by  one  quarter  inch;  between  5"  and  IS"  lengths  vary  by  one- 
half  inch;  above  IS"  lengths  vary  one  inch.  Any  ©length  of  bolt, 
however,  may  be  obtained,  but  odd  lengths  cost  more  than  the  atan 
dard  lengths.  By  length  of  bolt  is  meant  the  distance  from  point 
to  the  inner  side  of  head.  The  length  of  the  threaded  part  is 
from  three  to  four  times  the  height  of  nut.  Any  desired  length 
of  thread  may  be  obtained,  however. 


The  usual  forms  of  through  bolts  are  machine  bolts, 


rough  or  finished,  and.  carriage  bolts,  rough  (see  Fig. 11,  c). 

For  special  forms  see  catalogues.  In  general,  the  use  of  machine 
bolts  is  to  connect  iron  parts,  while  carriage  bolts  are  used 
in  wood  construction.  The  proportions  for  heads  and  nuts,  U.-.d, 
unfinished,  are  as  follows: 

Diameter  of  hexagonal  or  square  head  or  nut  across  plate 


= 1 1/2  d + 1/8” 


Height  of  nut  = d 

Height  of  head  = 3/4  d + l/l6" 


For  finished  heads  and  nuts  subtract  l/lG"  from  above  formulas. 

The  chamfer  on  heads  and  nuts  is  generally  drawn  as 


shown  in  Fig.  11  (a)  and.  (b), 


and  if  the  curves  are  drawn  with 
192 


■i 


. 

■ 


the  radii  given  t^ey  will  he  in  good  proportion. 


38 


(b)  Tap  Bolts,  more  frequently  called  cap  screws, 
do  nor  require  a nut,  but  screw  directly  into  one  of  the  pieces 
to  be  fastened,  the  head  pressing  against  the  other  piece. 

They  may  be  obtained  both  rough  and  finished,  and  with  hexagonal, 
square,  round  or  filister,  flat,  or  button  heads . Cap  screws 
are  threaded  either  U.S.S.or  V thread.  In  Fig.  12  are  illus- 
trated the  various  forms  of  heads  for  cap  screws.  In  this 
Fig.  (a)  represents  the  square  head,  (b)  the  hexagonal  head,  (c) 
and  (d)  the  filister  head,  (e)  the  flat  head  and  (f)  the  button 
head.  The  filister  head  may  be  obtained  either  flat  or  oval  on 
top.  All  cap  screws , except  those  with  filister  heads  are  thread- 
ed three-fourths  of  the  length  for  one  inch  or  less  in  diameter 
lengths  less  than  four  inches.  Beyond  these  dimensions  threads 
are  cut  one-half  the  length.  For  proportions  of  cap  screws, 
see  Kent,  p.  208.  Lengths  vary  by  one-quarter  inch  between 
the  limits  given,  and  height  of  head  except  in  the  flat  head 
is  equal  to  the  diameter  of  the  screw e The  angle  between  the 
sides  of  the  flat  head  is  7 6° . Radius  of  round  head  3/4  d, 

25,.  Set  Screws.  - These  a. re  screws  which  press  against 
a piece  and  by  friction  prevent  relative  motion  between  the  two 
parts.  They  are  usually  made  with  square  heads  and  case  hardened 
points,  and  may  be  obtained  with  either  U.3.G.  or  V threads. 
Usually  the  short  diameter  of  the  head  is  equal  to  the  diameter 
of  the  body  of  the  screw.  The  hight  of  head  is  always  equ  1 to 
diameter  of  body.  Lengths  vary  from  three-fourths  to  five  inches 

by  quarter  inches.  The  headless  set  screw  shown  in  Fig.  13  ( g) 

192 


■ 

: /,*  t ; 


. 


. . 


‘ 


. 


3/8" 


is  made  only  in  the  following  sizes; 


x 


«?  . 
5 


1/2"  x 9/lo" • 


5/8"  x ll/l6" r 3/4"  x 7/8" „ The  principal  distinguishing  feature 


of  set  screws  is  in  the  form  of  the  point.  Fig.  13  illustrates 
the  various  forms  of  points , Only  cup  and  round  point  set  screws 
are  regular,  ail  other  being  special.  In  dimensioning  set  screws 
give  the  diameter  first  and  length  last,  thus,  3/4"  x 8"  i.G. 

26,  Stud  Bolts.  - A stud,  is  a bolt  in  which  the  head 
is  replaced  by  a threaded  end  and  having  a small  plain  port.?  on 
in  the  middle  as  shown  in  Fig,  14.  It  passes  through  one  of  the 
parts  to  be  connected  and  is  screwed  into  the  other  parts,  thus 
remaining  always  in  position  when  the  parts  are  disconnected . 
with  this  construction  the  wear  and  crumbling  of  threads  in  a 
weak  material  such  as  cast  iron  is  avoided < Studs  are  generally 
used  to  secure  the  heads  of  cylinders  in  engine  design. 

There  is  no  standard  for  lengths  of  threaded  ends.,,  hence 
this  length  must,  always  be  specified.  They  may  be  obtained  mil- 
led at  B,  Fig,  14,  or  rough  and  the  ends  threaded  either 
or  V thread.  Lengths  vary  from  1 l/4"  to  8"  by  1-/4"  for  milled 
studs.  For  rough  studs  lengths  vary  from  1*  l/2"  to  4"  by  l/4"  and 
from  4"  to  6"  by  l/2"  . Usually  one  end  is  made  a tight  f it  T'hile 
the  other  is  standard. 

27.  Machine  Screws.,  Machine  screw  is  a term  used,  in 

/ l 

its  broad  sense,  to  Include  all  screws  that  fasten  into  iron  and 

metal  as  distinguished  from  those  screws  that  fasten  into  ood . 

We  shall,  however,  use  the  term  to  designate  those  screws  which  go 

by  gage  number  rather  than  by  diameter  of  the  body.  The  table 

Kent  p.  209  gives  all  the  proportions  which  are  required  in 

192 


• ' ■ 


• fc 


’ 


" 

■' 


' 

■i.  *i  mmM 


■ 


40 


drawing  such  screws*  It  will  be  observed  that  machine  screws 
have  no  standard  number  of  threads  per  inch,  hence  in  dimensioning 
these  screws,  give  the  number  of  the  screw,  the  number  of  threads 
and  the  length,  thus,  No.  30  - 16  x 1 l/f1'  hach.  Sc. 

08  «■ Forms  of  Screw  'Threads  , - The  different  forms  given 

to  screw  threads  aro  shown  in  Fig.  15.  Of  these,  the  United  states 
Standard  and  the  V threads  are  used  for  fastenings,  while  the 
Square,  Trapezoidal  and  Acme  threads  are  used  principally  for 
transmitting  motion. 

By  pitch  of  screw  is  meant  the  distance  from  a point  in 
one  thread  to  the  corresponding  point  in  the  next  thread.  Evident- 
ly the  number  of  threads  per  inch  of  length  is  equal  to  the 
reciprocal  of  the  pitch  for  a single  threaded  screw. 

When  a sciew  is  used  to  transmit  motion  it  is  often 

desirable  to  have  the  nut  advance  a considerable  distance  per 

revolution  of  the  screw.  It  is  evident  that  the  advance  of  the 

nut  along  the  axis  of  the  screw,  per  revolution,  is  equal  to  the 

pitch.  If  a considerable  advance  per  revolution  is  desired  it  may 

happen  that  the  pitch  necessary  will  be  too  great  for  tli •:  diameter 

of  the  screw.  This  difficulty  is  obviated  by  cutting  two  or  more 

parallel  threads,  each  having  the  same  pitch  or  lead.  Such  screws 

are  known  ao  multiple  threaded  screws;  when  a screw  has  two  threads 

it  is  called  a double  threaded  screw;  three  threads  a triple 

threaded  screw,  etc.  The  distance  between  two  consecutive  thro ads 

of  a multiple  threaded  screw  in  called  the  divided  pitch  and  is 

equal  to  tho  lead  divided  by  the  number  of  threads, 

192 


. 


' • " 


41 

Tlie  proportions  of  the  various  threads  shown  in  Fig. 
are  given  by  the  following  formulas: 

d = diameter  of  bolt  body 


d]_=  diameter  at  root  of  thread 
n = number  of  threads  per  inch 
p = pitch  of  screw 
t = depth  of  thread 

(a)  V Thread 


d^  = d - 


1 .733 


n 


tt  = 0 . 8f  6p  , . . < - 

p and  n same  as  (18)  and  (20). 

(b)  United  States  Standard  Thread. 


p = 0 c 24-\/d  + 0.685 
1.299 


0.175 


dl  -d  - 
1 

n=  i 

t = 0 . 65p 


n 


\ t t 


(17) 


(18) 

(19) 

(20) 
(21) 


(c)  Square  Thread  for  transmitting  motion.  (Wm.  Sellers  5 Co. 


ID  ia  . of 

| Screw 

1 

4 

5 

16 

rr 

o 

8 

7 

16 

1 

2' 

5 

8 

id 

4 

7 

8 

1 

1 

18 

1 

1Z 

td 

1q 

1 

In 

Threads 
per  In 

10 

9 

8 

7 

6 

1 

&2 

5 

4 1 

o 

A 

' O 

3 

n 

Dia.  of 

Screw 

- - - -- 

r— 

1— 

x8 

i! 

if 

2 

n 

2— 

cl 

Cjo 

ry 

3 

:.l 

*1 

r~r 

y 

■’z 

4 

i 

Threads 
per  in. 

2l 

o 

<5 

pi 

o 

k 

2* 

o 

Cj 

2 

r* 

ILL 

4 

1.2 

4 

i.2 

8 

ir> 

8 

li 

2 

li 

r> 

1192 


■ 


(d)  Trapezoidal  Thread  for  t rails  mi  t ting  mo  t i on . 


* 


P 

t 

( e ) Acme  Thread. 


2d 

15 

d 

io 


(22) 


(23) 


n 

- 1 

1 

ll  • 
2 

2 

4 

3 

1 

^ 1 

1 

5 

— 

6 

7 

8 

9 j 

10 

Si 

.3655 

2914 

2419 

.1801 

1451 

.1183 

! 

.087-5 

! 

r\  n n 

4 J w o 

.05  66 

.0470: 

.0411 

.0361 

.0319;- 

b 

.5707 

.2906 

2471 

1853 

.1483 

.1235 

.0927 

.0741 

.0618 

.0529 

.0463 

,041 5 

.0371 

t 

.5100 

.4100 

.3433 

.2600 

.2100 

.1767 

.1350 

.1100 

.0955 

.0814 

.07°  5 

.0855 

,n  goo 

In  using  the  above  formulas,  after  having  found  the 


pitch,  use  (20)  in  every  case  tofind  the  number  of  threads  per 

inch.  If  this  should  give  a number  other  than  some  convenient 

alch-quot  part,  the  pitch  should  be  changed  slightly  in  order  to 

give  the  convenient  number  of  threads.  A problem  will  illustrate 

this:  Suppose  it  is  desired  to  find  the  pitch  of  threads  on  a 

3"  bolt,  threads  to  be  U.S.Std.  Substituting  in  (18)  we  have 

p = 0.28"  and  from  (20)  n = 3.57.  Evidently  this  would  be  an 

inconvenient  number  of  threads  to  cut  on  the  lathe,  hence  suppose 

we  sfiy  n = 3.5,  from  which  p = 0.288". 

(f)  Gas  Pipe  Threads,  - The  rules  for  the  depth  and 

pitch  of  screw  threads  given  above  do  not  ap^ly  to  pipe  threads, 

since  the  calculated  depth  would  in  every  case  bo  greater  th  n the 

192 


* 


. 


• ' 


... 


. 


45 


thickness  of  the  pipe.  A section  of  a standard  pipe  thread  is 
shown  in  Fig.  15  ( g.) . It  will  bo  noticed  that  the  total  length  of 
thread  is  made  up  of  three  parts : full  thread  over  a tapered 

rj  p ?)  4.  4.  p 

length  of  — — — -----  9 when  D represents  t3=e  outside  diameter 

11 

of  pipe  and  n the  number  of  threads  per  inch;  two  threads  ful'i  at 
the  root  but  incomplete  at  the  top  and  not  on  a taper;  and  four 
imperfect  threads.  The  roots  of  these  last  four  threads  lie  on  a 
straight  line,  which  passes  from  full  depth  to  no  depth.  The 
total  ta'per  is  3/4"  per  foot.  For  number  of  threads  per  inch, 
outside  diameter,  inside  diameter,  weight  per  foot,  etc.,  see 
table  in  Kent,  pp.  194-5.  It  should  be  remembered  that  gas  pipe 
goes  only  by  inside  measurement,  i.e.  by  the  nominal  diameter. 

The  actual  inside  diameter  varies  somewhat  from  the  nominal,  but 
only  the  latter  is  used  in  speaking  of  commercial  sizes. 

59.  Hut  Locks.  Since  nuts  must  have  a small  clearance 
in  order  'to  allow  them  to  turn  freely  on  the  bolt,  the  tendency 

is  for  them  to  unscrew  or  slack  back.  This  tendency  is 

especially  true  in  the  case  of  nuts  subjected  to  vibration.  In 
order  to  prevent  this  unscrewing  a great  many  lifferont  arrange- 
ments have  been  devised. 

(a)  The  cheapest  and  most  common  device  is  the  loch 

or  jam  nut  shown  in  Fig.  16  (a).  Two  nuts  are  used,  ono  of  which 

is  about  half  as  thick  as  the  standard  nut.  Since  the  load  is 

thrown  on  the  outer  nut  (why1/),  that  nut  should  be  the  thicker. 

The  jam  but  is  not  always  to  be  depended  upon  as  a locking 

device.  For  proportions,  see  Fig.  1C  (a).  Very  often  the  size 

195 


- 


■ 


: 

' 


. 


■ 


, S, 


of  the  loci:  nut  is  the  same  as  the  standard. 

(b)  Another  effective  way  of  locking  nuts  is  by  means  of 
sot  screws,  as  shown  in  Fig.  16  (b).  This  arrangement  is  called 
a collar  nut.  The  lower  portion  of  the  nut  is  a cylinder  on  the 
surface  of  which  is  cut  a groove.  This  cylindrical  part  of  the 
nut  fits  in  a collar,  in  which  is  fastened  a dowel  pin,  as  shown. 

A set  screw  prevents  relative  motion  between  the  collar  and.  tho 


The  following  proportions 

have  proven  satisfaoto 

Height 

of  nut  above  ring 

5 * 

= Z d 

(24) 

Height 

of  ring 

= 0.55d 

(25) 

Ins ide 

diameter  of  ring 

= 1.5  d 

rr  If 

(26) 

Outside  diameter  of  ring  = 2.25d  — (27) 

16 

Diameter  of  cylindrical  parts  of  nuts  = 1.5d  and  1.45d  (28) 

»» 

*1 

Diameter  of  sot  screw  = 0.2d  + — — (29) 

16 

Diameter  of.  dowel-pin  = Q.ld  + 0.1"  ( '30 ) 

(c)  Fig.  17  (a)  shows  a device  for  locking  a nut  by 

means  of  a stop  plate  fastened  at  one  side  of  it.  Tho  plate  is 

bolted  to  one  of  the  pieces  to  be  secured  and  will  hold  the  nut 

in  either  of  two  positions,  or  in  other  words , the  nut  nay  be 

locked  at  intervals  of  one-twelfth  of  a revolution.  Frequently 
double  lock  plates  are  used  as  in  the  case  for  the  nuts  on  the 
studs  which  secure  the  propoller  blades  to  the  hub.  The  foil ow- 
ing proportions  may  be  used: 


192 


' 


.. 

. 


. . 


Thickness  of  plato  = —• 

A 1 

Diameter  of  cap  screw  = - _ 

. 4G 


(?1) 

(30) 


Distance  A = 1 l/8  times  the  short  diameter  of  nut  (33). 

(d)  A nut  lock  used  considerably  on  track  work  .1.3 
shown  in  Fig,  17  (b)  and  consists  essentially  of  one  complete 
turn  of  a helical  spring  placed,  between  tJ  e nut  and  the  piece  to 
bo  fastened  o When  the  nut  is  screwed  down  tightly  the  wash  ?r  is 
flattened  out  and  its  >.  lasticity  produces  a pressure  upon  the 
nut  thereby  preventing  it  from  s ladling  off. 

(e)  Very  frequently  split  p ins  are  used  to  prevent 
nuts  from  backing  off. 

^ (f ) When  using  fine  pitch  screws  nut  locks  arc  not 

always  necessary,  as  the  thread  angle  is  less  than  the  angle  .of 
friction.  A common  example  of  the  use  of  fine  threads  is  on  a 

bicycle . 


190 


A 


* 


CHAPTER  V 


46 


Keys  and  Cotters. 

50 - Keys,  - The  principal  function  of  keys  and  pins 
is  to  prevent  relative  rotary  motion  between  two  parts  of  a mach- 
ine, as  of  a pulley  about  a shaft  on  which  it  fits.  In  general 
keys  are  made  either  straight  or  slightly  tapering.  The  straight 
keys  are  to  be  preferred  since  they  will  not  disturb  the  alignment 
of  the  parts  to  be  keyed,  but  have  the  disadvantage  that  they  re- 
quire accurate  fitting  between  the  hub  and  shaft.  The  taper  keys 
by  taking  up  the  slight  play  between  hub  and  shaft  are  very  apt 
to  throw  them  out  of  alignment,  but  they  have  the  advantage  tret 
any  axial  motion  between  the  parts  is  prevented,  due  to  the 
wedging  action. 

Keys  may  be  divided  into  three  classes,  as  follows; 

(l)  sunk  keys;  (3)  keys  on  flats;  (3)  friction  keys. 

51.  Sunk  Keys.  - The  types  of  sunk  keys  used  chiefly 
are  those  having  rectangular  cross  sections,  though  occasionally 
round  or  pin  keys  are  used. 

(a)  Square  Key.  - The  so-called  square  key  is  only  ap- 
proximately square  in  cross  section  and  has  its  opposite  sides 
parallel.  As  shown  in  Pig.  18  (a)  this  type  of  key  bears  only  on 
the  sides  of  the  key  seats,  and  being  provided  with  a slight 
clearance  at  the  top  and  bottom,  it  has  no  tendency  to  exert 
a bursting  pressure  upon  the  hub.  To  prevent  axial  movement 

of  the  hub,  set  screws  bearing  upon  the  key,  or  other  mean/ 

192 


. 


must  be  provided.  The  square  key  la  used  where  accurate  con- 
centricity of  the  keyed  parts  is  required,  also  when  they  may 
have  to  be  disconnected  frequently,  as  in  machine  tools.  It  is 
suitable  for  heavy  loads  provided  set  screws  or  other  means 
are  used  to  prevent  tipping  in  its  seat. 

(b2  Flat  Key . - The  flat  key  has  parallel  sides,  but 

its  top  and  bottom  taper.  As  shown  in  Fig.  18  (b)  its  thickness 
is  considerably  loss  than  its  width  b,  and  that  it  fits  on  all 
sides  thus  tending  to  spring  the  connected  parts  atid  at  the 
same  time  introducing  a bursting  pressure  upon  the  hub.  This 
type  of  key  is  used  for  both  heavy  or  light  service  in  which 
the  objections  just  mentioned  are  permissible. 

(c)  Feather  or  spline.  - The  feather  key  or  spline 
is  a aquare  key  fitted  only  on  the  sides  and  permits  free  axial 
movement  of  the  hub  along  its  shaft.  Its  thickness  is  usually 
greater  than  its  width,  thereby  increasing  the  contact  surface 
and  at  the  same  time  decreasing  the  wear.  The  feather  key 

is  fastened  to  either  the  hub  or  shaft,  while  the  key-way  in 
the  other  part  is  made  a working  fit.  The  key  is  secured  to  the 
shaft  by  a countersunk  machine  screw,  or  when  it  is  required 
to  fasten  it  to  the  hub,  dovetailing  or  riveting  may  be  resorted 
to.  Quite  frequently  two  feather  keys  sot  180°  apart  are  used, 
thereby  equalizing  the  strain. 

( d ) 17 o odruff  Key,  - The  Woodruff  key  shown  i: 

18  (a)  is  a modified  form  .of  the  sunk  key.  It  is  patented 

and  is  manufactured  by  the  Whitney  llfg.  Co.  of  Hartford,  Conn, 

188 


The  key-seat  in  the  hub  is  of  the  usual  form,  but  that  in  the 
shaft  has  a circular  outline  and  is  considerably  deeper  than  the 
ordinary  key-way.  The  extra  depth  of  course  weakens  the  shaft, 
but  the  deep  base  of  the  key  precludes  all  possibility  of  tip- 
ping; and  the  freedom  of  the  key  to  adjust  itself  to  the  key 
seat  in  the  hub  makes  an  imperfect  fit  almost  impossible,  while 
with  the  ordinary  taper  key  a perfect  fit  is  very  difficult  to 
obtain.  In  securing  long  hubs  the  depth  of  the  key-way  may  bo 
diminished  by  using  two  or  more  Woodruff  keys  at  intervals  in 
the  same  key-seat.  This  form  of  key  may  be  obtained  in  both 
the  square  and  flat  types . 

(e)  Lev/ is  Key.  - Some  twenty  years  ago  Mr.  Wilford 
Lewis  invented  the  type  of  key  shown  in  Fig.  19  (b).  This  key 
is  subjected  practically  to  a pure  compression,  and  is  used  ex- 
tensively by  one  manufacturer  on  large  engine  shafts.  Its  main 
disadvantage  lies  in  the  fact  that  it  is  expensive  to  fit. 

(f)  Barth  Key.  - Several  years  ago  Mr.  C.  G.  Barth 
invented  a key  shown  in  Fig.  SO  (a).  It  consists  of  an  ordinary 
restangular  key  with  one  half  of  both  sides  beveled  off  at  45°  .• 

It  is  not  necessary  that  the  key  forms  a tight  fit  since  the 
pressure  tends  to  force  it  better  into  its  seat.  This  key  has 

no  tendency  whatever  to  turn  in  its  seat,  and  is  subjected  to 
a pure  compression.  In  that  respect  it  is  similar  to  the  Lewis 
key  and  has  the  advantage  that  it  costs  less  to  fit.  This  vey 
may  also  be  used  as  a feather  key,  and  in  a great  many  cases 
has  replaced  troublesome  rectangular  feather  keys  and  has  always 
proved  satisfactory. 


192 


' 


• • • 


- 

* 


■ 

AO 

(g)  Round  or  Pin  Key.  ~ A round  or  pin  key  ferns  a 
cheap  and  accurate  means  of  securing  a hub  to  the  end  of  a shaft. 
This  form  of  fastening  is  used  only  for  light  and  si^all  work.  - 
The  pin,  either  cylindrical  or  tapering,  is  fitted  half  way  into 
the  shaft  and  hub,  as  shown  in  Fig.  20  (b).  Sometimes  a screw 
is  employed  in  place  of  the  pin.  When  using  a taper  pin  it  is 
advisable  to  use  "standard  taper  pins",  as  those  may  be  purchased 
at  less  cost  than  they  can  be  made  in  small  lots.  Reamers  to  fit 
these  various  sizes  may  be  obtained  from  any  machinist  suppy  com- 
pany, 

52,  Keys  on  Flats.  - A key  on  the  flat  has  par. ilel 
sides  with  its  top  and  bottom  slightly  tapering  and  • is  used  for 
transmitting  light  powers . Fig.  21  (a)  shows  this  form  or  fasten- 
ing . 

55.  Friction  Keys.  - The  most  common  .form  of  friction 
key  is  the  saddle  key  shown  in  Fig.  21  fb) . The  sides  are  paral- 
lel arid  its  top  and  bottom  are  slightly  tapering.  The  bottom  fits 
the  shaft  and  the  holding  power  of  the  key  is  that  due  to  friction 
alone.  This  form  of  key  is  intended  for  very  light  work,  or  in 
some  cases  for  temporary  service  as  in  setting  an  eccentric, 

54 . Strength  of  Keys.  - Keys  are  generally  proportioned 

by  empirical  formulas  and  tables,  and  in  almost  all  cases  these 

are  based  upon  the  diameter  of  the  shaft,  neither  the  twisting 

moment  on  the  shaft  nor  the  length  of  the  key  are  consider  mi. in 

arriving  at  the.  cross  section.  Since  a key  is  used  for  tori iof* 

alone  the  twisting  moment  to  be  transmitted  should  fix  its 

102 


* 


. 


' 


• 

■ 


50 


dimensions,  and  not  the  diameter  of  the  shaft,  as  in  most  cases 
the  shaft  must  also  re  ist  a bonding  moment  in  addition  to  the 
twisting  moment,  thus  requiring  a shaft  of  larger  diameter  than 
is  necessary  for  simple  twisting.  This  means  that  the  empirical 
formulas  give  a larger  hey  than  is  really  needed,  thereby  in- 
creasing its  cost  and  at  the  same  time  decreasing  the  effective 
strength  of  the  shaft.  The  length  of  thbjcey  should  bo  considered 
to  determine  its  crushing  and  shearing  resistance. 

Generally  the  twisting  moment  to  be  transmitted  by 
the  hey  is  determined  by  some  machine  element  other  than  the 
shaft,  as,  for  example,  a gear  or  pulley  on  the  shaft. 

In  calculating  the  dimensions  of  the  hey,  the  size  of 
shaft  hould  not  be  disregarded  altogether  or  the  result  might 
be  a key  too  small  to  be  fitted  properly,  or  one  that  is  too 
large.  Prom  this  we  arrive  at  the  following  method  of  procedure | 
calculate  the  required  dimensions  and  modify  these  to  suit 
practical  considerations , 

It  is  generally  supposed  that  hoys  fail  by  ’hearing 
across,  but  this  is.  seldom  the  case.  A large  number  of  f llsr's 
arc  due  to  the  crushing  of  the  side  of  the  hey  or  key-seat,  and 
it  is  always  well  to  determine  the  crushing  stress. 

(a)  Crushing  Strength,  - To  determine  the  crushing 
stress  in  the  side  of  any  key-seat,  the  following  method  may  be 
used  (See  Pig.  22 ). 

Lot  P = driving  force  exerted  by  the  hub 

S-D=  crushing  fiber  stress 

192 


I 


. 

. 

' 

• 

v 

. 

* 


51 


Ss=  shearing  fiber  stress 
T = twisting  moment  to  be  transmitted 
1 = length  of  key. 

The  crushing  resistance  of  the  key  is 


tlS 


™ and  its 


moment  about  the  center  of  the  shaft  is  approximately 


tld£ 


A 


Equating  this  moment  to  T,  we  have 

O,  = 4T_ 

"b  t Id 


(34) 


Assuming  S-^  and  having  given  values  for  T,  t and  d, 
(34)  may  be  used  for  calculating  the  required  length  of  the  key. 
Occasionally  a key  is  required  to  transmit  the  full 
power  of  the  shaft;  hence  making  its  strength  equal  to  that  of 
the  shaft,  we  get 

5 

(1X  d S B t 

• 16  = — 

from  which  t = ' > Ss 


b 


4t S|~j  ( 35 ) 

(b ) fi ho a r i n g S t r e n g t h , - To  determine  the  shearing  stress 
equate  the  moment  T to  the  product  of  the  radius  of  the  shaft  and 


the  stress  over  the  area  exposed  to  shear,  whence 


(36) 


s bid 

Equating  the  value  of  T from  (34)  to  that  obtained 
from  (36)  we  get 

2 S„b 


t = 


(37) 


5b 


192 


' 


• 

. 


If  S-.,  = 2S  , as  is  generally  assumed,  (57)  calls  for  a 
3 

square  key.  To  facilitate  fitting,  the  width  of  the  key  is  very 
often  made  greater  than  its  depth,  which  has  the  effect  of  de- 
creasing S relative  to  S,  . From  this  it  follows  that  invest! ga~ 
tions  for  the  crushing  stress  are  more  essential  than  those  for 
the  shearing  stress,  as  the  latter  in  an  actual  key  ta'-'os  care  of 
itself . 


55.  Gib-head  key.  - The  gib-head  key.  Fig.  .35.  (b)  is 
exactly  the  same  as  the  flat  key  shown  in  Fig.  18  (b)  with  the 
head  added,  and  is  used  when  it  cannot  be  driven  out  from  the 
small  end  conveniently.  It  is  variously  called  gib-head  key,  hoo 
head  key,  nose  key  and  draw  key.  Proportions  of  the  head  are 
shown  in  the  figure. 

56  o Goiters.  --  A cotter  is  a cross  key  used  for  jo  ini 
rods  and  hubs  that  are  subjected  to  a tension  or  compression  in 
the  direction  of  their  axis,  as  in  a piston  rod  and  cross  head; 
a strap  and  connecting  rod,  valve  rod  and  sterna  Fig.  03  shows 
one  method  of  joining  two  parts  by  means  of  a cotter. 

Strength  and  Proportions.  - (1)  Assuming  the  joint  in 
Fig.  23  to  be  loaded  axially  as  shown,  the  following  relations 
between  the  external  force  and  the  internal  strength  at  the  va- 
rious sections  may  readily  be  obtained. 

Let  S>b  = allowable  bearing  or  crushing  stress 
fl  shearing  stress 

" tensile  stress. 


SB  = 


St  = 


(a)  For  tension  in  rods 
0 _ TTd2  St 


(58) 


192 


end  across  the  slot 


^ <9 


(b)  For  tension  in  the  rod 


p = 


dxt 


st 


L 4 

(o)  For  tension  in  the  s ocliet  across  the  slot 
2 


P = | — ( D 

L4 

(0..;  For  shearing  the  cotter 


dr)  - (D  - )t 


J 


O JL. 
L 


(59) 


( 40 ) 


= 2btS, 


(4-1) 


/ \ 
<,  e / 


For  shearing  the  rod  end 


P = 2 ad^Sg  (42; 

(f)  For  shearing  the  socket  end 

P = 2c  (D  - d1)  Ss  (41) 

(g)  For  crushing  along  the  surface  AB 

P = ditSfc  (A4) 

(h)  For  crushing  along  the  surface  CE  and  FGf 

P = (D  - dp)  tSb  (45) 


(2)  Assuming  the  load  on  the  joint  to  be  reversed  in 
direction  so  that  the  rods  are  in  compression  instead  of  tension, 


'.76  then  have 

(i)  For  shearing  the  collar  off  the  rod 

F = ^dieSs  (4?) 


(j) 

For  crushing 

the 

collar 

■Q  yf  2 

P = — ( 9-0  - 

4 

■ dx 

Sx  * 

) -g 

(47) 

The  taper  on 

the 

cotter 

should  not  be 

j: lade  ex c e o s i v e 

so  as  to 

permit  loosening 

due  to 

the  load  t ran e 3 

■;i  ittod.  0 "c  a s i on 

ally  set  screws  are  used  to  guard  against  this  loosening  of  the 

192 


54 

cotter,  A practical  taper  is  1/2  inch  per  foot,  but  thi"  m ay 
be  increased  to  1 l/2  inch  per  foot  if  some  locking  device  is 
applied  to  the  cotter. 


In  the  figure  the  cotter  is  shown  as  being  square 
ended,  but  more  often  it  is  made  semi-circular,  which  has  the 
following  advantages:  (l)  avoids  sharp  corners  that  arc  liable 
to  start  cracks | (2)  gives  more  shearing  area  oh  the  sides  of 
the  slots;  (3)  cheaper  to  make  the  slots. 


Exercise.  - Deduce  empirical  formula  for  a,  b,  c,  dn , dn 


D,  e,  and  t in  terms  of  d,  assuming  Sg  = 0,8S 

192 


t 


' i ..  kD  ^ 

b s 


CHAPTER  VI 


o a 


Shafting* 

Shafting  is  of  three  kinds,  namely,  rough,  turned  and 
cold-rolled,  and  is  made  almost  exclusively  of  steel,  wrought 
iron  having  gone  out  of  use  to  a large  extent.  The  rough  shafting 
is  turned  only  at  journals  and  couplings,  and  is  very  little  iced. 
The  t urne d.  s h af t in g is  used  extensively  for  accurate  work , 
and  the  standard  sizes  run  l/lo  of  an  inch  under  the  quarter  inch- 
sizes:  this  is  due  to  the  fact  that  the  regular  sizes  of  rough 
shafting  are  turned  down  that  amount.  Cold  rolled  shafting  has 
a very  tough  crust  on  its  surface,  acquired  by  its  treat  -nt  in 
manufacture,  and  for  that  reason  from  20 y.  to  50  y stronger  than 
turned  shafting  of  the  same  size.  It  may  he  obtained  in  almost 
any  size  and  in  cheaper  than  turned  shafting,  hut  the  variation 
in  diameter,  which  is  one  of  its  characteristics , prohibits  its 


57 o Shaft  Calculations 


M 1 


the  following  two  important  points  must  be  considered:  (1 ) for 
short  or  very  large  shafts  the  strength  determines  their  size  * 


(?)  for  long  shafts  stiffness  and  rigidity  generally  fix  the 
diameter  to  be  used* 

The  general  principles  governing  straining  actions  to 
rtinj  ay  be  subjected,  are  as  follows:  (a)  Simple  twist- 
ing: (b)  Simple  bending  : (c)  C or 'bine  d twisting  and  bending:  (d) 

Combined  twisting  and  compression. 

19? 


Strength  - Shafting  is  very 


58 , Simple  Twisting.  - 
rarely  subjected  to  simple  twisting,  since  the  weights  of  pulleys 
and  gears,  belt  pulls  and  gear  tooth  pressure  cause  bending  stresses. 
These  stresses  are  quite  frequently  difficult  to  determine  before- 
hand, and  as  they  are  apt  to  complicate  calculations  they  are 
omitted  in  many  cases  * Whenever  they  are  omitted  a large  factor 
of  safetj-  is  used  in  order  to  tal:e  care  of  the  bending. 

Let  d = diameter  of  shaft 

P = load  acting  at  the  end  of  lever  arm 


S = allowable  shearing  stress*; 


Equating  the  moment  of  the  load  to  the  moment  of  resistance, 

Pa  = V-  d°Ss  = ? 

Tra- 


il 


O] 


16 


(d?) 


S 


s 


TC  d° 

Taole  I gives  the  values  of  for  different-  diameter 


16 


of  shafts,  and  — = constant  in  the  table. 


Problem.  - Find  the  diameter  of  a shaft  which  is  to 
sustain  a twisting  moment  of  80,000  inch  pounds.  Fiber  stre^ ' nc 
to  exceed  18000  pounds  per  square  inch. 

E =6.67.  The  nearest  constant  in  Table  I above  3-*  37 

C* 

8 

is  6.75.  The  corresponding  diameter  is  3 \/'-y  which  is  the 
diameter  of  the  required  shaft. 


Angular  Deflection 


The  relation  between  the  twistir 


CD 


moment  7 and  the  angular  deflection  x may  be  derived  as  follow; 

108 


. 


■ 


- 


. 

> 

57 


Lot  = torsional  modulus  of  elasticity. 

1 = length  of  shaft  in  inches. 

x = deflection  in  inches  measured  on  the  -urfe.oe  of 
the  shaft  shown  in  Figa  25  „ 

TABLi]  I . 


d 

0 

4 

r- 

1 

"i 

± 

3 

1 

o 

n 

/ 

n 

1 

16 

8 

16 

4 

16 

8 

16 

o 

(-U 

1 

±198 

, 235 

,279 

. . 32S 

.363 

.443 

,510 

r,  c : R 

0 ...  Is—  *_/ 

f 

; c 6C1 j 

9 

1 o 57 

1 72 

1 o 881 

2 c 05 

2,23 

2 .44 

2, 63 

2 . 34 

3,05 

3 

50  29 

5 „ 63 

5 c,  98 

6 c 35 

6 a 73 

7.13 

7,54 

7.937 

8.41 

4 

12  o 55 

13  o 15 

13,77 

14 . 40 

15.-  05 

15,73 

16.42 

17. .14 

17.87 

R 

24  e 51 

25  c 44 

26  040 

27  « 36 

28,37 

29,41 

30  - 45 

31,53 

32.62 

' 

6 

4 2 o 3 5 

43  * 69 

45  r 06 

43  , 95 

47  c 87 

49./S3 

50.80 

52.32 

R r Or 

' c Jo 

7 

67  o 25 

69  3 08 

70o  92 

72  c 81 

. . 

74,72 

, , - - 

76,68 

78  o 65 

80 . 60 

82.72 

8 

100.39 

102  o 77 

105.17 

107  e 62 

110,10 

i 

112,63 

115,16 

1 ■ 1 

117,73 

120*41 

±563 


' 

> 


- . . 


o 

1C 

CD|Ol 

11 

'16' 

ic. 



1 

! 

Si  5 | 

l 

_ i 

7 

O 

o 

15 

16 

j 

d 

i 

! 

1 

i 

• i 

.84" 

. 945 

1.05 

1,17 

1,29 

1 . 43 

1 . 

. 4 . 

3.30 

rr  r- 

O o oO 

3.81 

• 

4.08 

4.37 

4.66 

4.97 

1 

2 

i 

P Q7 

O • o / 

Q rr  a 

a „ 

- _ , 

9.84 

, ...  . - . , 

10.34- 

IQ.  87 

T 

11.41  11.98 

I 

' 

3 i 

1 

L 

18.  S3 

19,40 

20.20 

21,01 

. ... 

21,86 

22.72 

O ^ /3 

wO  « *w; 

4 ! 

■'.'3.76 

34.90 

' 

36.09 

37.28 

38.52 



39.76 

41.05 

. 

! 

5 j 

i 

55.48 

57.01 

- 

58.65 

60  , 30 

62.06 

63, 7g 

65.48 

5 i 

84.82 

86.95 

89,10 

91  o 27 

93 51 

op,  na 

'.J  <i  1 \J 

98.07 

. . 

i 

7 

. . 

123.11 

125.81 

r 

l 

1128.58 

| 

131 c 36 

134.21 

137.07 

140,00 

i 

8 | 

Then 
Length  of 


x 

1 ~ ' 

arc  x 


Et 

TTd _9 

330 


Substituting  this  in  (49) 

SoO  1 S 

9 = 

red  Et 


(so) 


Substituting  in  (50)  the  value  of  S obtained  from  (48),  we  have 


to  one 


6 = 584  1 T 

d4'  Et 

The  angle  6 in  common  practice  is  generall 

degree  in  twenty  diameters  in  length  of  shaft. 

Exercise.  - Calculate  the  allowable  fiber  s 

192 


(51) 

y limited 

tresses  for 


59 


steel  and  wrought  iron  shafts  so  that  the  angular  deflection  is 
kept  within  the  limits  given  above,  assuming  for  steel  Z^  = 
15,000,000,  for  wrought  iron  = 11,000,000. 

59  f>  Simple  Bending  - Strength  - 'In  designing  machinery 
we  frequent ly  use  stationary  shafts,  upon  which  certain  heavy 
parts  revolve,  these  revolving  pieces  being  brass  bushed.  Such 
shafts  are  sometimes  called  pins,  and  may  be  proportioned  for  a 
simple  bending  moment,  since  it  is  not  required  to  transmit  any 
twisting  moment. 

Equating  the  bending  moment  M to  the  moment  of  resis- 


tance, we  have 


M = 


7Td‘ 


32 


or 


M 


=0.1  d°  (nearly) 


! n;o  Ni 
V Od  ) 


Table  II  gives  the  values  of  0.1  for  different 

M 

diameters  of  shafts,  and  3 = the  constant  in  the  table. 

192 


TABLE  II 


60 


|d 

0 

1 

16 

I 

s 

rr 

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■ 


■ 


. ■ 


. 


61 


Problem*  - Find,  the  diameter  of  a shaft  vfhich  is  to 
take  a bending  moment  of  45*000  inch  pounds.  Fiber  stress  not 
to  exceed  15*000  pounds  per  square  inch* 

M = Za  The  nearest  constant  in  Table  II  above  5 is 

S 

5.05,  corresponding  to  a diameter  of  5 1/8",  which  is  the  required 

diameter . 

Transverse  Seflection.  - Transverse  deflection  is  that 
due  to  the  bending  of  the  shaft  and  may  occur  alone  or  in  con- 
nection with  the  angular  deflection  discussed  in  Art.  (58).  For 
line  and  counter  shafts  a deflection  of  0.01  of  an  inch  per  foot 
of  length  is  considered  good  practice,  while  for  certain  kinds  of 
machinery  this  would  be  excessive.  Transverse  deflection  depends 
upon  the  method  of  loading  the  shaft,  and  formulas  for  these  deflec- 
tions may  be  obtained  from  the  corresponding  beam  formulas  given 
in  Kent „ 

40.  Combined  Twisting  and  Bending*  - A rotating  shaft 
carrying  gears,  pulleys,  drums,  etc.,  is  subjected  to  both  bending 
and  twisting,  when  used  for  transmitting  power.  Calculating  the 
diameter  of  the  shaft  by  either  of  the  above  tables,  ignoring  the 
other,  would  result  in  a weak  shaft*  We  may,  however,  substitute 
for  the  simple  twisting  moment  T a -greater  twisting  moment  T0, 
which  is  the  equivalent  of  the  combined  twisting  moment  T and  the 
bending  moment  M. 

Thi3  equivalent  twisting  moment  is  derived  direc  ly  from 
the  expression  for  combined  stress,  as  follows: 

Merriman  in  his  Mechanics  of  Materials,  p.  265,  gives  the 

following  formula  for  combined  stress: 

192 


■ 


62 


3 ‘ / 2 -g2 

Max*- '"ten  • . or~T*-omp0  S"t:r&-su  Se  - -g*  +y  Ss^  + '•% 


(53) 


Substituting  in  (53)  the  values  of  Sa  and . S.  from  (48) 
and  (52)  above 


^0 


srrd3 

is 


16 

Tdl 


\ 


\L2 


M + /T"  + M' 


2 


(54) 


n» 


- m\/  t2  + 


K-ow  since  — — equals  the  equivalent  twisting  moment  Te 

M2  ( 55 ) 

To  find  the  diameter  of  the  shaft  suitable  for  the 
combined  moments  T and  M,  substitute  the  value  of  Te  for  T in 
(48)  and  proceed  as  outlined  in  Art.  38. 

Equation  (55)  is  the  one  used  almost  altogether  by 


American  designers. 

Guests  * Law  - In  1800  Prof.  J.  J.  Guest  published  an 
article  in  the  Philosophical  Magazine  giving  results  of  a series 
of  experimental  researches  on  the  strength  of  ductile  materials 
under  combined  stress.  He  found  that  the  shearing  stress  is  the 
factor  which  governs  the  yielding  of  the  material  when  subjected 
to  combined  stress.  From  this  theory  he  deduced  the  following 
formula  for  the  equivalent  bending  moment i 

Me  =\/~m2  + T2  (56) 

In  England  (56)'  is  now  adopted,  as  the  correct  formula 
to  be  used  in  the  design  of  shafting,  since  it'  is  based  upon  ac- 
tual experiments.  It  will,  also  be  considered  as  the  standard 
formula  to  be  used  in  the  solution  of  problems  pertaining  to 
this  course. 

To  find  the  diameter  of  a shaft  subjected  to  comb'ned- 

bending  and  torsion,  calculate  the  equivalent  bending  moment 

192 


••  \ 

• 

' 

i!  L 

■ 

' 

■ 

’ 


by  means  of  (56)  and  substitute  it  for  M in  (52)  and  proceed  as 
outlined  in  Art.  39. 

Problem,,-  Find  the  diameter  of  a shaft  subjected  to 
a twisting  moment  of  80,050  inch  pounds  and  a bending  moment  of 
4-5000  inch  pounds.  Fiber  stress  not  to  exceed  12,000  pounds  per 

square  inch. 

From  (56)  Mg  = 91780,  and  from  (52)  —A? — = 7.65. 

The  nearest  constant  in  Table  II  above  7„35  is  7.67,  corres- 
ponding to  a diameter  of  4-  l/4”  , which  is  the  required  diameter. 

To  show  the  difference  in  the  results  obtained  by  unin 
(55)  we  find  the  diameter  of  shaft  to  be  3 7/8” , thus  showing 
that  it  is  better  to  use  (53). 

By  comparing  the  three  problems  just  shown,  one  readil 
sees  the  importance  of  considering  both  the  bending  and  the  twin- 
ing moments  upon  any  shaft  that  is  subject  to  these  actions. 

4:1.  Combined.  Twisting  and  Compression.  - Propeller 
shafts  of  steamers  and  vertical  shafts  carrying  considerable 
weights  are  subjected  to  these  straining  actions.  Two  cases 
arise  in  practice,  as  follows: 

(a)  Then  the  span,  or  distance,  between  bearings  is  so  smal 
tfcat  the  shaft  may  be  considered  as  subjected  to  simple  compres- 
sion, so  far  as  the  action  of  the  thrust  is  concerned. 

(b)  '.Then  the  span  is  so  great  that  the  shaft  must  be  consi- 
dered as  a column  liable  to  buckle. 


Case 

( a ) - The 

intensity 

of 

compressive  stress  fci  a 

solid  shaft  is 

AP 

Q — 

0 7rn  ’ 

in  which 
192 

p 

equals  the  thrust.  From 

. 

, 


(48)  the  intensity  of  shearing  stress  due  to  the  twisting  moment 

16  T 


on  the  shaft  is  S. 


■ 7T  d': 


’he  resultant  maximum  stress  due 


to  the  combined  action  of  Sc  and  Sg  may  be  found  from  the  expres’ 
sion  for  combined  stress,  see  (58) „ / 


Max,  Compressive  Stress  is 


\L  2 , 
+ v os 


s 2 

bc 


(t7) 


Substituting  values  of  SQ  and  Ss  in  (57),  we  have 

o j j ^4-  1 

Max  o Comp0  Stress  = ^ ^ 2 j P * \/  P * up  j ( 58) 

To  find  d for  given  values  of  F,  f and  maximum  compres- 
sive stress,  assume  a trial  value  of  d (somewhat  larger  than 
required  for  twisting  moment  alone),  and  then  check  for  maximum- 

stress  „ 


Case  (b)  - The  value  of  mean  intensity  of  compressive 
stress  in  a long  column  is  given  by  Hitter’s  formula  (p.  711, 
Meriman’s  Mechanics  of  ‘Materials),  that  is 


H TC"~hr 


in  which  P = the  thrust  or  load 
L = unbraced  length 
A = area  of  cross  section 


E = coefficient  of  elasticity 

S = unit  stress  at  the  elastic  limit 

S’=  the  greatest  compressive  unit  stress  on  the 
concave  side, 
m = varies  from  .25  to  4. 

= least  radius  of  gyration  of  cross  section. 
172 


r 


/ 


Gb 

Since  the  stress  found  by  (59)  is  the  re an  intensity 

of  compress j ve  stress  in  the  long  column  which  corresponds  to 

a maximum  compressive  stress  S0,  a short  compression  member  of 

the  same  cross  section  A would  be  capable  of  supporting  a load 

which  is  greater  than  P in  the  ratio  of  to  3’  or  the  new  load 

c c 

C p 

pt  = ug  (do) 

a i 
C 

Plow  to  find  the  diameter  of  shaft,  necessary  to  sup- 
port a load  P and  twisting  moment  T,  use  (58)  as  before,  but 
substitute  P'  for  P» 


42 Hollow  Shafts  c - In  any  shaft  the  outer  fibers  are 
mere  useful  in  resisting  bending  or  twisting  than  the  fibers  near 
the  center,  from  which  it  follows  that  the  weight  of  a ho 1 1 ow 
shaft  is  diminished  in  greater  proportion  than  its  strength. 

Let  d = the  diameter  of  a solid  shaft  having  the  sane 
strength  as  the  hollow  shaft. 


d-j_  = outer  diameter  of  hollow  shaft, 
dp  = inner  " " " ” 


For  a hollow  shaft  subjected  to  simple  twisting 

m _ £s^fdf  - af) 

X ’ ...  ■ - ■ ■■  - M - - — ■ - 

IS  dx 

The  value  of  T for  a solid  shaft  is  given  by  (48) 
Henoe  to  make  the  two  shafts  equally  strong,  we  must  have 


(51) 


13  d: 


1 99 


s 

/ 


IS 


•r 

i 


ee 


or  d1"  = d- 


d0  a 


i - { » ) 


Letting 


d- 


= m 


di  = 


\ l m*' 

V ~ 1 


d \ /— 


For  average  value  of  m = 2 


(62) 


( 


) 


d1  = 1.02d.  ( 64 )_ 

45.  Bending  Moments,  - In  calculating  the  bending 
moment  that  a shaft  is  subjected  to,  the  various  authorities 
differ  in  the  method  used  for  determining  the  length  of  the  mo- 
ment arms.  It  is  generally  assumed,  by  all,  that  a shaft  running 
freely  in  its  bearings  is  so  loose  that  it  could  easily  deflect  to 
the  center  of  the  bearing.  Therefore  for  a shaft  of  this  kind  the 
moment  arms  are  measured  to  the  centers  of  the  bearings,  and  a 
shaft  designed  on  this  assumption  is  generally  on  the  safe  side  as 
far  as  strength  is  concerned. 

Whenever  a machine  part  is  driven  or  forced  tightly 
upon  a shaft,  it  is  practically  impossible  for  the  shaft  to  break 
at  the  center  of  the  hub,  but  may  fail  near  either  end.  This  is 
due  to  the  fact  that  the  hub  may  become  loose  since  the  bending  of 
the  shaft  would  tend  to  localize  the  crushing  at  those  points. 
According  to  Mr.  C.  L.  Griffin  this  distance  may  be  assumed  to 


lie  between  l/2  inch  and  1 inch. 

The  method  of  assuming  the  moment  arms  as  extenGin  * to 

the  center  of  the  hubs  and  bearing  is  the  one  coma 'only  used,  and 

192 


, 


67 

-.Till  be  adopted  in  all  problems  in  this  course.  It  is  simpler 
and  saves  considerable  time, 

44.,  Problems.  - The  following  problems  are  given  to 
illustrate  the  general  method  of  procedure  in  obtaining  the 
bending  moment  diagrams  and  finally  determining  the  size  of 
shaft  required  in  any  given  case. 

(a)  Two  Gears  between  Bearings.  - Let  Fig.  23  show 
the  method  of  loading.  1-Tow  calculating  the  reactions  at  A and 

B due  to  the  load  on  the  gear  e,  we  can  readily  find  the  bending 
moments  at  the  points  G.  In  like  manner  we  can  determine  the 
bending  moments  at  D due  to  the  load  on  the  pinion  f . TTow  con- 
struct the  bending  moment  diagrams  ASB  and  AFB  as  shown,  thus 
showing  at  a glance  where  the  maximum  moment  occurs,  and  at  the 
same  time  permitting  of  a graphical  addition.  Besides  sustaining 
the  bending  moments  the  shaft  also  transmits  a twisting  moment 
T = PR.  To  obtain  the  required  diameter  find  the  value  of  M 
and  proceed  as  explained  in  Art.  40, 

(b)  Gear  and  Two  Pinions,  - The  loading  of  the  gear 
and  pinions  is  shown  in  Fig.  27  (a).  Proceeding  as  outlined 
above  lay  off  to  some  convenient  scale  the  bending  moment  dia- 
grams for  each  load,  and  since  P and  W act  in  the  same  direction, 
the  bending  moments  due  to  these  loads  will  oppose  each  other. 
Thus  the  shaded  portion  of  the  diagram  represents  the  algebraic 

sum  of  the  two  moment  diagrams.  How  combining  the  twisting- 
pis 

moment  T = — — with  the  maximum  bending  moment  as  outlined  in 

•Zj 

Art.  40,  the  required  diameter  of  the  shaft  may  be  obtained. 


' 


, 


68 

Assuming  the  loads  P and  U acting  at  so  s angle  to  each 
other.  In  order  to  find  the  bending  moment  at  any  point  on  the 
shaft  as  at  D,  lay  off  separately  in  the  direction  of  each  force, 
the  bending  moment  ordinate  for  that  force.  For  example,  at  the 
point  C the  moment  M-j  due  to  the  loads  W is  BF  in  Fig.  87  (a), 
and  the  moment  Mg  due  to  P is  CG«  Assuming  the  loads  acting  at 
right  angles  to  each  other,  lay  off  in  Fig.  87  (b)  Mn  parallel  to 
W and  Mg  parallel  to  P,  their  resultant  M gives  the  bonding 
moment  at  the  point  C.  Having  found  the  maximum  moment  by  this 
method,  and  knowing  the  twisting  moment,  find  the  necessary  dia- 
meter of  the  shaft  as  outlined  above. 

Note.  - The  methods  described  in  the  above  problems 
may  be  applied  to  ail  forms  -of  leading. 

(o)  Hoisting  Drum.  - Fig.  88  shows  a common  method  of 
supporting  the  hoisting  drum  of  a crane  trolley.  The  drum  is 
hushed  and  runs  loose  on  the  shaft,  hence  no  twisting  moment  is 
transmitted  through  the  shaft. 

Let  G - the  weight  of  the  gear. 

D = " ,f  " 11  drum. . 

? = " lohd  oh  the  gear  teeth  acting  horizontally, 

* W = " ?f  " each  rope . 

Reducing  all  load.s  to  the  more  heavy  load  support  we  have 
for  the  horizontal  lead  at  A 

H = J (1  - t)  (87) 

and  for  the  vortical  load  at  A 


192 


> 


V = Gr(  1 


t)  + W(b  + 2c 0 + Dd 


(3?) 

. £ 

Combining  H and  V in  a manner  -similar  to  that  used 
in  Prob.  (h)  for  moments,  the  resultant  pressure  at  A is  obtained* 

By  similar  reasoning  the  reaction  and  resultant  pres- 
sure at  B may  be  obtained* 

Knowing  the  resultant  pressures  at  A and  B calculate 
the  bending  moments  at  C and  S,  and  whichever  is  the  maximum 
must  be  used  in  finding  the  required  size  of  the  shaft,  using  the 
method  given  in  Art.  39. 

(d)  In  Fig.  29  is  shown  diagrammatic ally  a hoisting 
drum,  gear  and  pinion,  in  which  E is  the  center  of  gravity  of 
the  rope  loads  W.  Whenever  there  are  two  ropes  on  the  drum, 
the  position  of  E is  constant  while  for  one  rope  E moves  along 
the  drum,  and  for  the  latter  case  several  solutions  should  be  made 
with  varying  positions  of  E, 

The  load  W is  supported  upon  throe  points,  namely,  the 
journals  A and  B and  the  gear  tooth  at  D„  The  load  W taken 
separately,  causes  an  upward  reaction  on  each  journal,  and.  Is 
divided  proportionally  between  them.  The  tooth  load  P Is  also 
divided,  proportionally  between  A and  E,  and  causes  a downward 
reaction  on  each  journal.  The  algebraic  sum  of  the  two  loads 
upon  the  journals  gives  the  amount  and.  direction  of  the  resultant 
load  on  the  journal. 

In  Fig.  29  draw  BD  and  DE  produced  to  cut  AB  at  C.  We 
thus  represent  the  rope  load  W as  supported  eccentrically  upon  a 
beam  BD  with  an  arm  EH  and  prevented  from  rotating  about  3D  by 

1 *T  OO 

1 *7*3 


I 


■ 


’ 


70 


the  reaction  of  the  bearing  upon  the  journal  A acting 
HG.  From  this  we  derive  the  following  relations: 

A downward  pressure  of  the  bearing  upon  the  journal  A 

An  upward  " " " " " " " E 


with  an  arm 


EF 


■W=W? 


FA 

FD 

BD 


(W  + TV  ) 


m f? 


BF 

on  the  gear  teeth  at  D = grj 


(W  + Wf ) . 


The  condition  of  loading  at  the  journal  A is  seen  from, 
the  position  of  the  point  0,  which  lying  beyond  B as  in  Fig.  2r- , 
indicates  a downward  pressure  upon  A;  a position  at  B would  indi 
cate  no  pressure  upon  A,  and  a position  between  A and  B would 
indicate  an  upward  pressure  on  the  journal.  In  the  above  dis- 
cussion the  weight  of  the  drum  and  gear  was  neglected. 

192 


1 


CHAPTER  VII* 


71 


Bearings  and  Journalej 

45.  Bearings.  - Bearings  may  be  divider?,  into  two  general 
classes:  (l)  Sliding;  (2)  Rolling* 

Sliding  Bea’rin-g.  - There  are  three  types  of  sliding 

bearings : 

(a)  Right  line  bearings  in  which  the  motion  is  parallel 
to  the  elements  of  the  sliding  surfaces:  This  type  may  again  be 
subdivided  into  three  hinds  as  follows:  (1)  Ordinary  flat  guides 
a,s  used  on  engine  crossheads  and  punching  machine  rams.  (2) 

Angular  guides  as  used  on  planers  and  lathes.  (5)  Circular  guides 
as  used  on  some  engine  crossheads,  piston  and  stuffing  boxes. 

(b)  Journal  bearings.  - When  two  machine  parts  rotate 
relatively  to  each  other,  the  part  which  is  enclosed  by  and  rubs 
against  the  other  is  called  the  journal , while  the  part  which  en- 
closes the  journal  is  called  the  box  or  less  specifically  the 
bearing.  Journal  bearings  may  be  divided  into  three  classes:  (i) 
Journal  rotating  inside  a fixed  bearing,  for  example  a shaft  in 
its  bearing.  (2)  Journal  fixed  and  bearing  rotating  as  a loose  « 
pulley  on  its  shaft.  (3)  Journal  and  shaft  both  having  a definite 
motion  as  in  the  crank  end  of  a connecting  rod, 

(c)  Thrust  bearings.  - A thrust  bearing  is  one  designed 
for  taking  the  end  thrust  of  a shaft.  There  are  two  kinds:  (l) 

Step  or  pivot  bearing  is  used  on  vertical  shafts  and  supports  the 
weight  of  the  entire  shaft  together  with  its  attached  parts,  and 
any  other  force  acting  vertically  downwards . The  shaft  terminates 
at  the  bearing. 

192 


' 


•in-  ■ >1 j ;• 


72 


(2)  Collar  or  thrust  bearing  as  used  on  a propeller  shaft  or  on  a 
spindle  of  a drill  press.  In  this  case  the  shaft  extends  through 
and  beyond  the  bearing. 

Rolling  Bearings.  - Rolling  bearings  may  be  divided  into 
two  types  as  follows:  (a)  Ball  bearings,  in  which  hardened  steel 
balls  are  placed  between  the  journal  and  its  box,  thereby  reducing 
frictional  resistance,  since  sliding  friction  is  changed  to  rolling 
friction.  (b)  Roller  bearings  using  either  cylindrical  or  conical 
rollers  in  place  of  the  balls.  This  arrangement  distributes  the 
load  over  a larger  surface. 

Each  one  of  the  above  types  maybe  subdivided  into  the 
following:  (l)  Right  line  bearings;  (2)  Journal  bearings:  (?) 

Thrust  bearings. 

46.  Bearing  Surfaces  for  Journals.  - Bearing  surfaces 
are  made  of  many  different  substances,  depending  largely  on  the 
class  of  work  in  which  the  bearing  is  used,  and  the  conditions 
under  which  it  must  run.  The  following  is  a list  of  some  of  the 
materials  that  are  used  for  bearing  surfaces:  phosphor  bronze, 
ordinary  bronze,  lumen  bronze,  babbit,  cast  iron,  chilled  cast 
iron,  tempered  steel,  case-hardened  steel,  mild  steel,  also  the 
self-lubricating  materials,  fiber-graphite  stone  and  wood. 

The  materials  phosphor  bronze,  ordinary  bronze,  lumen 

bronze,  and  babbitt  may  be  classed  as  anti-friction  metals,  and 

together  with  other  alloys  of  a similar  character,  are  U3ed  for 

lining  bearings  on  machinery  of  rather  high  grade.  One  of  the 

objects  of  these  linings  jj_?s  to  prevent  wear  on  the  journal,  by 

192 


running  it  in  contact  with  a metal  softer  than  itself.  This 
result  is  very  desirable  because  the  bee  ring  metal  cen  be  much- 
more  eaoi  iy  and  cheaply  replaced  than  the  journal. 

The  main  requirements  for  a good  bearing  metal  are  the 
following:  (&)  Least  liability  to  heating.  (b)  Sufficient  strength 

#*r 

to  prevent  squeezing  out  under  the  load.  (c)  High  melting  point. 

(d)  Least  abrasion  in  service.  (o)  Least  possible  abrasion  of 
j ournal . 

Experimental  Conclusions.  Some  years  ago  the  Pennsyl- 
vania Railroad  made  3one  exhaustive  service  teste  with  various 
combinations  of  cop.-ar,  tin  and  lead,  in  order  to  determine  the 
composition  which  would  be  best  suited  to  thsir  requirements, 
and  the  following  conclusions  were  drawn  from  the  experiments: 

(a)  Ordinary  bronze  shows  50%  more  wear  then  phosrher 
bronze . 

(b)  The  phosphorus  plays  mo  part  in  preventing  wear,  save  by 
producing  sound  castings. 

(c)  Wear  diminishes  with  tne  increase  of  lead. 

(d)  Wear  diminishes  with  d^mi’nution  of  tin. 

(e)  Alloys  containing  more  than  15-/?  of  lead  or  loss  than 

8;":  of  tin  could  not  be  produced  because  of  segregation:  but  it  is 

believed  that  if  the  lead  coulch  be  still  farther  increased- and 

the  tin  decreased  and  still  ha>ve  the  resultant  alloy  homogeneous, 

a,  better  metal  would  result. 

Since  that  time  allojye  have  been  successfully  forced 

containing  a much  larger  percentage  of  lead, 

192 


and  tests  b av  e 


I 


'*3 

* 


74 


born©  out  th©  conclusion  of  the  Pennsylvania  Railroad.  How- 
ever,  the  gain  in  ivearing  qualities  is  somewhat  off?  et  by  the 
decreased  compressive  strength  of  the  metal. 

Leadj  - Of  all  bearing  metals  lead  is  by  f tr  the  first 
in  anti-friction  qualities,  but  it  has  so  little  compressive 
strength  that  many  alloys  containing  large  percentag  s of  it 
will  yield  under  ordinary  pressure,  with  the  result  hat  the  metal 
is  squeezed  into  the  oil  inlets  and  stops  lubricatio:  .. 

Bronzes . - The  composition  of  phosphor  br  nze  and  ordi- 
nary bronze  ae  used  by  the  Penn.  R.  R.  , is  as  folio  s:  Phosphor 
bronze  contains  79.7 y'o  copper,  10 fi  tin,  9.5^  lead  and  .8$  phos- 
phorus; Ordinary  bronze  87.5^  copper  and  12.5^  tin. 

Lumen  bronze,  invented  by  Prof.  R.  C.  Carp  nter  of 
Cornell  Univ . and  manufactured  by  the  Bierbaum  and  Merrick  Metal 
Co.  of  Buffalo,  is  claimed  to  be  an  excellent  substi  ute  for 
phosphor  bronze.  It  has  a high  compressive  strength  and  pos- 
sesses the  peculiar  property  when  heated  of  incr9asi:  g in  strength 
until  a temperature  of  350®  F.  is  reached. 

Babbitt  Metal.  - Babbitt  metal  is  probably  the  ?->ost 
extensively  used  of  all  the  bearing  metals,  one  roan  n for  this 
being  that  the  term  is  applied  to  an  innumerable  numb < r of  dif- 
ferent white  metal  alloys.  A good  composition  is  as  follows: 
lead  70^,  antimony  20;'?,  and  tin  10fe. 

For  further  information  regarding  bearing  rr  stal  alloys 
see  Kent  pp.  319  to  358. 

The  physical  condition  and  structure  often  ias  as 

192 


, 

. 


.. 


i.  *-;*•$  *n  " I 

■ 

- 


. 

♦ 

• 

■ 


• . 


‘ J) 


75 


much  to  do  with  the  running  qualities  of  a bearing  motrl  as 
the  chemical  composition.  This  may  be  determined  by  a micro- 
scopic examination  of  a fractured  section  which  has  been  polished 
and  etched  with  acid.  The  microstructure  test  should  be  included 
in  specifications  for  bearing  metals  in  important  work.  The 
conditions  which  cause  heating  of  the  bearing  and  should  be 
particularly  avoided  are  : (a)  Segregation  of  the  metal:  (b) 

Coars'e  crystal  lino  structure;  (c)  Dross  or  oxidation  products 
and  an  excessive  amount  of  enclosed  gas  in  the  metal. 

The  materials  cast  iron,  chilled  cast  iron  and  mild 
steel  ar--  often  used  in  cheap  work.  Cast  iron  bearings  with 
steel  journals  have  met  with  considerable  success  when  the  bearings 
were  long,  and  some  eminent  engineers  have  advocated  their  exten- 
sive use,  claiming  that  the  surface  will  in  a short  time  wear  to 
a glassy  finish  and  run  with  very  little  friction.  However,  if 
for  any  reason  lubrication  fails  and  heating  begins,  the  result 
is  liable  to  be  either  serious  injury  or  total  destruction  to 
both  bearing  and  shaft,  and  for  that  reason  the  cast  iron  bearing 
has  fallen  into  disfavor.  Tempered  steel  and  case-hardened 
steel  have  been  used  rather  extensively  for  bearings  where  the 
pressures  are  great. 

Self-lubricating  materials  are  used  in  places  whore 

i 

lubrication  would  be  very  difficult  and  liable  to  be  neglected 

or  where  oil  or  grease  would  be  harmful.  Fiber-graphite  ie  a 

patent  substance,  and  is  composed  of  pulverized  graphite  mixed 

with  wood  pulp  in  a bath,  of  water.  The  mixture  is 

192 


then  moulded 


. 


.. 


' 


, 


' 


* 


■ 

'' 


. 


. 


• 

. 

76 

to  the  desired  shape,  and  the  water  is  squeezed  out  by  heavy  -pres- 
sure, after  which  it  is  treated  with  oil  and  baked.  It  is  said 
to  work  very  well  under  ordinary  pressures. 

Stone  is  very  little  used  for  bearings  in  this  coun- 
try, except  for  light  work  when  minimum  fric'tion'is  desir- 
ed, in  which  case  the  precious  stones  are  used  extensively.  The 
jewels  of  watches,  of  electrical  instruments,  furnish  excellent 
examples  of  this  class  of  bearings. 

Wooden  bearings  are  frequently  used  when  the  bearing 
is  submerged  in  water,  and  are  also  used  occasionally  in  cheap 
machinery. . The  stern  tube  bearing  for  the  propeller  shaft  of 
a ship  furnishes  a good  example  of  a bearing  submerged  in  water, 
and  is  made  of  lignum  vitae,  a wood  which  is  well  adapted  for 
service  of  this  kind. 

47.  Lubrication.  -,The  function  of  the  lubricating 
device  is  to  keep  up  a continuous  film  of  oil  between  the  journal 
and  its  bearing,  so  that  the  metals  will  never  be  in  actual 
contact.  When  this  is  done  the  journal  practice. lly  floats  in  a 
bath  of  oil,  and  runs  with  very  little  friction.  However,  when 
lubrication  fails  the  coefficient  of  friction  is  at  once,  increased, 
the  bearing  begins  to  heat  and  the  journal  either  seizes  or  cuts 
out  the  bearing  metal  very  rapidly. 

The  lubricants  used  for  different  kinds  of  bearings 
range  from  the  thinnest  oils  to  heavy  grease,  and  in  some  cases 
even  to  graphite,  a solid.  For  slow  speeds,  with  heavy  contin- 
uous pressures  grease  i3  better  than  oil,  because  it  is  not  so 

192 


77 

4 

easily  squeezed  out  from  between  the  surface  of  contact:  while 
with  faster  speeds  the  oil  is  continuously  carried  to  the  point 
of’  maximum  pressure  so  rapidly  that  there  is  not  time  enough 
for  the  oil  film  to  be  squeezed  out.  In  general  it  may  bo  said 
that  the  viscosity  or  "body"  of  the  lubricant  varies  inversely 
as  the  speed  of  the  journal,  and  directly  as  the  pressure  upon  it. 
Usually  the  oil  reservoir  is  a part  of  the  bearing 
itself,  but  when  the  bearing  is  not  a casting  it  is  more  conven-  • 
ient  to  have  the  oil  cups  separate. 

It  is  now  quite  common  to  oil  the  bearings  of  engines 
in  large  power  plants  by  forced  lubrication.  The  oil  is  forced 

* j 

by  means  of  a pump  through  various  bearings,  after  which  it  is 
returned  to  the  reservoir,  filtered,  and  again  forced  through 
the  system.  In  this  way  the  bearings  are  oiled  continuously, 
and  also  cheaply,  because  the  oil  is  used  over  and  over  again 
and  needs  very  little  replenishing, 

46.  Design  of  Journals.  - In  designing  a journal  there 
are  usually  three  things  which  must  be  considered;  first,  the 
journal  must  be  strong  enough  as  a beam  to  support  the  load  which 
comes  upon  it,  and  if  it  is  also  subjected  to  a twisting  moment 
this  must  be  combined  with  the  flexure  in  calculating  the  strength 
second,  it  must  be  sufficiently  rigid  to  obviate  springing  and 
consequent  heating  of  the  bearing;  and  third,  the  pressure  per 
square  inch  of  projected  area  must  be  Iovt  enough  to  prevent 

squeezing  the  lubricant  from  between  the  bearing  surfaces. 

192 


. 

* 


' 


78 


(a)  Strength.  - End  journals  are  generally  considered 

cantilever  beams  loaded  uniformly,  (soe  Fig.  30).  Equating  the 

bending  moment  to  the  moment  of  resistance, 

PI  _ 7?  d3S 
d 32 


from  which 


d = 1.7 


v ; p 1 

si 


(77) 


(b)  Stiffness.  - From  the  table  of  beams  in  Kent’s 


p.  238,. the  deflection 


A = P1'3 

(en) 

p-pT 

W-h  -A- 

In  good  practice 

the  valu  ) of  A is 

generally  limited 

to  — - — 

of  an  inch. 

100 

Substituting  in  ( 

■38)  this  value  of 

A and  for  I its 

equivalent 

in  terms  of  the 

diameter  we  have 

approximately 

4 AS 

d = 4 \ i1  1 

(69) 

’ E 

Allowable  Bearing  Presaun  e.  ~ The  following  table, 


taken  from  Unwins,  and  other  authori  ties,  shows  the  allowable 
pressure  per  square  inch  of  projects  l area  for  different  types 


of  machines. 

(a)  Bearings  for  slow  speed.e  an  L intermittent  load.s 

Main  journal  on  punching  m .chines  2000  - 3000 

Grant:  pins  on  punching  mac!  ines  5000  - 80n0 

(b)  Engine  crank  pins 

High  speed  engine  250  - 600 

192 


Low  speed  engine 

859 

— 

1300 

Locomotive 

1500 

- 

1700 

Marine 

400 

- 

000 

Petrol 

350 

- 

400 

(c) 

Engine  cross  head  pins 

High  speed  .engine 

soo 

- 

1700 

Low  ,f  H 

1000 

- 

1800 

Marine 

1000 

~ 

1300 

Petrol 

soo 

-!■ 

in  no 

(a) 

Engine  main  journals 

Center  crank  high  speed  engine 

100 

- 

240 

Side  u low  " f( 

100 

?G0 

Petrol 

550 

- 

400 

Marine 

150 

400 

(e) 

Locomotive  driving  journals 

Passenger 

190 

Fr  e i ght 

GOO 

Switching 

G20 

(?) 

Car  journals 

300 

600 

(g) 

Motors  and  generators 

50 

- 

90 

(h) 

Thrust  blocks  for  propeller  shafts 

40 

- 

80 

(i) 

Engine  slides 

Marine 

50 

- 

ICO 

Stationary 

o r 

fj  s..* 

- 

40 

(3) 

Eccentric  3heaves 

so 

_ 

100 

(k)  Hoisting  machinery  shafting 

192 


GO 


. 

so 


The  length  of  a journalln  proportion  to  the  diameter 
is  a ratio  which  the  designer  must  first  choose  according  to  his 
own  judgment,  and  afterwards  adjust  to  the  three  conditions  men- 
tioned above.  The  ratio  varies  from  .5  to  5 or  6,  usually  ranging 
from  2 to  3 for  ordinary  rigid  bearings  when  there  is  no  limitation 
on  space,  from  3 to  4 for  rigid  shaft  bearings,  and  from  4 to  5 
for  self-aligning  shaft  bearings.  In  general,  the  ratio  decreases 
as  the  diameter  increases  and  increases  as  the  speed  increases. 

49.  Design  of  Bearing.  - Having  determined  the  dimensions 
of  the  journal,  the  next  step  is  to  design  the  bearing.  This  is 
an  excellent  example  of  the  kind  of  machine  design,  where  calcu- 
lations are  of  little  value.  Should  an  exact  analysis  of  the 
stresses  be  made,  it  would  be  found  that  they  arc  very  complicated, 
and  in  many  cases  so  small  that  the  proportions  of  the  part 
would  be  determined  by  other  considerations.  For  this  reason 
it  is  impracticable  to  calculate  the  thickness  of  walls  and  gen- 
eral proportions  of  the  bearing,  and  they  are  usually  taken  from 
parts  which  have  been  in  actual  service  and  have  been  found 
successful  under  conditions  similar  to  those  of  the  new  design, 
scale  drawings  are  often  contained  in  manufacturers T catalogs 
with  a table  of  the  principal  dimensions  for  different  sizes,  and 
these  are  a great  help  to  inexperienced  designers,  A designer, 
however , should  make  himself  independent  of  those  aids  bv  stud v- 
ing  standard  designs,  and  becoming  so  familiar  with  t-'em  that  ho 
is  able  to  proportion  parts  of  this  kind  simply  from  judgment. 

In  addition  to  these  pi pfybrtions  there  are  a few  impor- 
tant points  of  design  with  which  he  should  be  thoroughly  familiar. 

192 


. 

I 


. 


y- 

u 

■ 


«r 

• 

(1)  Some  means  of  adjustment  for  wear  should  be  provided. 
This  adjustment  may  be  made  in  various  ways  but  the  most  com- 
mon methoe  is  to  use  a split  bearing  bolted  together,  in  which 
case  the  wear  may  be  taken  up  by  simply  tightening  the  bolts  as 
shown  in  Fig.  31. 

(2)  The  line  of  division  of  the  bearing  should  bo  perpenuicU' 
lar  to  the  line  of  pressure  so  that  the  surface  having  maximum 
pressure  on  it  will  be  continuous  and  not  have  any  sharp  edges 

as  they  would  tend  to  scrape  off  the  oil  film.  The  division 
should  be  made  as  shown  in  Fig.  31  for  two  reasons:  first,  be- 
cause when  made  in  that  way  it  helps  to  prevent  the  box  under 
pressure  from  springing  together  at  the  3id.es  and  gripping  the 
shaft:  second,  because  it  prevents  the  lubricant  from  escaping 

at  the  split. 

(3)  Eearings  should  always  fit  loosely  at  the  points  where 
they  are  divided,  since  the  sharp  edge  would  tend  to  scrape  the 
oil  film  off  the  shaft  if  it  were  tight,  and  also  because  it 

is  the  natural  tendency  of  a box  which  has  been  bored  absolutely 
true  to  fit  tightly  at  the  3ides,  and  more  especially  so  if  it 
begins  to  beat.  This  action  is  illustrated  by  Fig.  32. 

(4)  The  method  of  applying  the  lining  to  a bearing  is  some- 
what different  for  the  different  metals.  The  brasses  and  hard 
alloy  linings  are  cast  separate,  and  when  used  in  split  bearings 
are  aquare  or  some  irregular  shape  to  keep  them  from  turning 

in  the  bearing.  Boxes  made  in  this  manner  are  hand  fitted,  but 

when  they  are  used  in  a solid  box  it  is  customary  to  turn  the 

192 


■ 

* 


4 


. 

* 

. 

’ 


■ 


* 


bushings  on  the  outside,  bore  the  bearing  and  then  drive  the 
bushing  to  place. 

The  babbitt  linings  are  formed  by  pouring  the  molten 
metal  around  a mandrel  the  same  size  as  the  journal,  into  a 
recess  in  the  bearing  which  was  provided,  f'cr  that  purpose. 

After  the  babbitt  cools  the  shaft  may  be  readily  withdrawn,  In 
better  work  the  babbitt  is  poured  around  a mandrel. of  smaller 
diameter,  and  then  hammered  or  peened  in  and  boned  out  to  size. 

(5)  In  a bearing  of  any  considerable  length  the  pressure 
between  the  journal  and  the  box  is  not  uniformly  distributed 
along  the  entire  length  and  the  maximum  pressure  naturally  occurs 
at  the  point  where  the  box  is  rigidly  held.  This  point  is  usually 
located  in  the  center  of  the  bearing,  so  for  this  reason  the 
oiling  device  on  a cylindrical  journal  should  be  at  that  point  if 
possible.  The  lubricant  will  naturally  move  in  the  direction  of 
least  resistance  and  then  find,  its  way  to  the  ends  of  the  bear- 
ing;1 at  ?vrhich  points,  either  wipers  or  drip  pans  should,  be  Pro- 
vided so  that  the  oil  cannot  run  out  on  the  shaft  and.  drip  to 

the  floor. 

(6)  Then  it  is  impracticable  on  account  of  the  length  of 
a bearing,  or  its  construction,  to  place  the  oiling  device  at 

the  center,  the  proper  distribution  may  be  accomplished  by  the 
use  of  oil  grooves  in  the  bearing  metal.  Oil  grooves  may  bo 
divided  into  two  classes:  first,  the  grooves  proper  which  distri- 
bute the  oil  in  the  bearing:  and  second,  the  -feeders  which  carry 

the  oil  to  the  grooves.  The  oil  grooves  should  always  be  cut 

192 


■* 


■ 


. 


■ 


. 


■ 


RPW 

' 


. 

■ 


- 

. $pi 


parallel  to  the  journal,  and  be  placed  at  a point  where  the 
prespure  is  slight,  so  that  they  wi'  1 serve  to  fcr^  a uniform 
and  continuous  film  of  oil  on  the  3 urnal , and  not  act  as  a wiper 
to  scrape  off  what  oil  film  ha3  air  )ady  been  formed.  The  feeders 
should  always  be  placed  at  right  ar  ^les  to  the  journal.  Although 
the  groove  is  often  cut  obliquely  : t is  bad  practice  as  the 
oblique  edges  are  more  injurious  t<  the  film  than  straight  ones, 
and  they  cannot  be  placed  a,t  point  of  least  pressure. 

(7)  ho  escape  should  be  provided  for  the  oil  at  points 
of  maximum  pressure. 

50.  Design  of  Ca;^  and  I sits . ~ The  oar  of  a bearing 
subjected  to  an  upward  pressure  if  generally  regarded  as  a beam 
supported  by  the  holding  down  bol  s or  screws  and  loaded  at  the 
center.  Both  its  strength  and  st  .ffness  should  be  considered. 

Let  b = distance  between  bo?  is 

e = depth  of  cap  as  she  n in  Fig.  33 
1 = length  of  bearing 
P = maximum  upward  pres  jure 

£ = maximum,  allowable  6 ^flection  = ”rgf“  of  an  inch. 

Strength  - Equating  the  ber  ting  moment  to  the  moment  of 
resistance 

Pb  _ Sle^ 

A 


from  which 


o 


V 


84 


Stiffness  -From  the  table  of  beams  in  Kent* 3 p.  808,  the 
deflection 


4S  - 


Pb' 


48  El 


Substituting  in  (71), 

le' 


100 


(71) 


of  an  inch  for  & f 18,0PQ,P00 


for  E (cast  iron),  and  for  I,  then 

18 

e = .0112b 


(72) 


Bolts  or  Screws.  - The  holding  down  bolts,  screws  or  studs 
should  bo  designed  for  tension,  assuming  two  thirds  of  the  max- 
imum upward  pressure  comes  upon  one  bolt. 

192 


t 


- -■  iig 


' 


CHAPTER  VI 1 I 


Spur  Gearing. 

Spur  gearing  may  be  of  tvo  forms , friction  gearing  and 
toothed  gearing. 

51.  Friction  Gearing.  - The  simplest  form  of  gearing:  is 
the  plain  friction,  consisting  of  a pair  of  cylinders  or  cones 
hold  in  contact  with  sufficient  pressure  to  produce  rotation  of  the 
driven  gear  simply  by  the  friction  between  the  two  surfaces.  (See 
the  bevel  frictions  on  the  shaper  in  the  University  Wood  Shop.) 

These  gears  aro  used  when  the  speed  of  rotation  is  high,  and 
when  an  absolutely  positive  drive  is  either  non-essential  or  un- 
desirable. In  friction  drives  the  driving  wheel  is  often  made-  of 
paper  of  come  sort  of  fiber,  while  the  driven  wheel  is  made  of 
metal.  However,  in  many  cases  both  gears  ar ? made  of  metal.  The 
velocity  ratio  of  a pair  of  frictions,  provided  there  is  no  slip- 
ping between  the  surfaces  of  contact,  is 

D,  Ho 

— = — (73) 

Jo 

in  ~*hich  D and  IT  are  the  diameters  ond  revolutions  per  minute  res- 
pectively. The  subscript  1 refers  to  the  driver  and  2 to  the  driven, 
V.'ith  this  type  of  gearing  there  is  always  a certain  amount  of  slip- 
page which  must  be  taken  into  consideration,  and  which  with  properly 
designed  gears  ^ hould  be  about  2:/ , 

5P.  Toothed  Gearing.  - Then  it  is  desired  to  transmit  an 

absolutely  positive  velocity  ratio,  or  when  the  surface  speed  is  not 

19£ 


very  great,  it  becomes  necessary  to  provide  the  surface  with 
projections  and  grooves  as  shown  in  Fig.  f>4.  The  original  sur- 
faces of  the  frictions  then  become  the  pitch  surfaces  of  the 
toothed  gears,  and  the  projections  together  with  the  grooves  form 
the  teeth.  These  teeth  must  be  of  such  a f err.  as  to  satisfy  the 
following  conditions: 

(a)  They  must  transmit  a uniform  velocity  ratio.  In 
order  to  do  this  the  common  normal  at  the  point  of  contact  of  the 
tooth  profiles  must  always  pass  through  the  pitch  point,  i.e.  the 
point  of  tangency  of  the  two  pitch  lines. 

(b)  The  relative  motion  of  one  tooth  upon  the  other 
should  be  as  much  a rolling  motion  as  possible,  on  account  of  the 
greater  friction  and  wear  attendant  to  sliding.  With  toothed 
gearing,  however,  it  is  impossible  to  have  pure  rolling  contact 
and  still  maintain  a constant  velocity  ratio. 

(c)  The  tooth  should  conform  as  nearly  as  possible  to 

c cantilever  beam  of  uniform  strength,  and  should  be  symmetrical 
on  both  sides  so  that  the  gear  may  run  in  either  direction, 

(d)  The  arc  of  action  should  be  rather  long,  so  that 
more  than  one  pair  of  teeth  may  be  in  mesh  at  the  same  time. 

Tooth  Curves.  - There  are  a great  many  different  curves 
that  would  serve  as  profiles  for  teeth  and  satisfy  the  above  con- 
ditions with  sufficient  accuracy  for  all  practical  purposes,  but 
the  ones  in  actual  general  use  are  only  two,  namely,  the  involute 
and  the  cycloidal . As  regards  strength  and  efficiency,  the  two 

forms  aro  practically  on  a par.  However,  the  involute  tooth  has 

192 


I 

. 


orr 


one  decided  advantage  over  the  cycloidal;  namely,  the  distance 
"between  centers  may  be  slightly  greater  or  less  then  the  theoreti- 
cal distance,  without  affecting  the  velocity  ratio.  The  cycloi- 
dal tooth,  also,  has  one  important  advantage  over  the  involute; 
namely,  a convex  surface  surface  in  always  in  contact  with  o,  concave 
and  although  the  contact  is  theoretically  a line,  practically  it  is 
net,  and  consequently  the  wear  is  not  so  rapid  as  ’■•ith  involute 
teeth  when  the  surfaces  are  all  convex. 

55.  Methods  of  Manufacture.  - Gear  teeth  are  formed  in 
practice  by  two  distinct  processes:  moulding  acid  machine  cut  ■:  ing . 
Originally  all  gears  were  cast  and  the  mould  was  formed  from  a 
complete  pattern  of  the  gear.  Of  late  years,  however,  gear  mould- 
ing machines  have  been  used  to  a considerable  extent  end  the  re- 
sults obtained  are  far  superior  to  the  pattern  moulded  gear.  Even 
with  machine  moulding  , however,  the  teeth  are  somewhat  rough 
and  warped  out  of  shape,  so  that  the  gears  always  run  with  con- 
siderable friction  and  are  not  suited  to  high  speeds.  In  this 
country  the  gears  of  ordinary  size  s/re  almost  always  cut,  except 
in  cheap  machinery.  The  method  which  is  most  commonly  used  is  to 
cut  them  with  a milling  cutter,  which  has  been  formed  to  the  ex- 
act shape  of  the  tooth.  There  are  also  two  stylos  of  gear  planers, 
one  of  which  generates  mathematically  correct  profiles  by  virtue 
of  the  motion  given  to  the  cutter  and  blank,  and  the  other  forms  the 
outlines  by  following  a previously  shaped  templet. 

54.  Materials  of  Gearing.  - The  material  used  for  gear 

teeth  are  machine  steel,  steel  casting,  cast  iron,  bronze,  rawhide, 

192 


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I 


. 


. 


. 

. 

. I 


86 


fiber  and  wood.  Machine  steel  pinions  are  often  used  with  large 
cast  iron  gears  in  order  to  he  up  for  the  weehre00  of  the  teeth 
on  the  pinion,  due  to  their  decreased  section  at  the  root,  by  using 
a strong  material.  Steel  castings  are  used  when  the  gears  are 
of  large  size  and  are  subjected  to  violent  shocks  and  heavy  loads. 

Bronze  is  frequently  used  for  srur  uinions  meshing 
with  steel  or  iron  gears,  and  when  properly  cut  ^ay  be  run  at  very 
high  speeds.  In  worm  gearing  the  gear  is  often  made  of  bronze 
and  the  worm  of  steel.  Good  bronze  is  considerably  stronger  than 
cast  iron. 

Cast  iron  is  the  material  which  is  more  frequently 
used  by  far  than  any  other. 

Rawhide  and  fiber  gears  are  used  when  quiet  and  smooth 
running  free  from  vibrations  is  desirable.  Rawhide  gears  are 
stronger  and  preferable  to  fiber.  The  Hew  Process  Rawhide  Co.  of 
Syracuse,  N.Y.  claim  their  gears  to  be  equally  as  strong  as  cast 
iron  gear  of  the  same  dimensions.  They  are  furnished  with  or 
’without  metal  flanges  and  bushings,  and  the  teeth  are  cut  just  the 
same  as  the  metal  gear.  They  are  usually  of  smell  size,  although 
larger  ones  are  sometimes  made  in  which  the  teeth  only  are  of 
rawhide,  the  center  being  made  of  cast  iron.  They  are  often  used 
as  the  driving  pinions  on  motors,  and  the  fact  that  rawhide  is  a 
non-conductor  is  a marked  advantage  in  this  instance. 

Wooden  teeth  are  sometimes  used  for  the  large  gear  of 

a pair,  in  which  case  the  teeth  of  the  mating  gear  a.r^  usually  cast. 

The  wooden  teeth  or  cogs  are  morticed  into  a cast  iron  rim.  and 

1S2 


their  purpose  is  the  same  as  in  the  c&ae  of  the  rev  hide  and  liber 
gears,  to  reduce  the  noise  and  vibration  by  their  greater  elac- 

f 

ticity . 

55.  Involute  System,  - In  the  involute  system  ef  veer- 
ing the  outline  of  the  tooth  in  an  involute  of  a circle  called 
the  base  circle.  However,  when  the  tooth  extends  b'dow  the  bc.se 
circle  that  portion  of  the  profile  is  made  radial.  The  simplest 
conception  of  an  involute  is  as  follows:  if  a cord,  which  ha" 
been  previously  wound  around  any  given  plane  curve,  and  has  a 
pencil  attached  to  its  free  end,  is  unwound,  keeping  the  cord 
perfectly  tight,  the  pencil  will  trace  the  involute  of  the  given 
curve.  The  base  circle  may  easily  be  obtained  by  'bring  through 
the  pitch  point  a line  making  an  angle  with  the  tangent  to  1:'  e 
pitch  circle  at  this  point,  equal  to  the  angle  of  obliquity  of 
action;  then  the  circle  drawn  tangent  to  this  line  will  be  the 
required  base  circle. 

In  order  to  manufacture  .gears  economically  it  is  very 
essential  that  any  gear  of  a given  pitch  should  work  correctly 
with  any  other  gear  of  the  same  pitch,  thus  making  an  interchange' 
able  set.  To  accomplish  this  end  standard  proportions  have  b ?sn 
adopted  for  the  teeth. 

The  angle  of  obliquity  of  action  which  is  generally 
accepted  as  standard  in  this  country  is  15°,  although  in  oases 
of  special  design  this  angle  is  often  made  greater,  and  is  some- 
times as  large  as  30°.  When  the  angle  of  obliquity  is  increased, 
the  component  of  the  pressure  tending  to  force  the  gears  awart 

and  producing  friction  in  the  bearings  is  of  course  increased, 

192 


■ 


. 

. 

v , ^ 

H «< 


. 


but  on  the  other  hand  the  profile  of  the  tooth  *Hd?r 

at  the  base  and  consequently  the  strength  is  correspondingly 
greater.  These  special  gears  are  used  when' the  conditions  are 
unusual  and' the  standard  tooth  form  is  not  suitable.  In  England 
teeth  of  greater  obliquity  of  action  and  less  depth  then  -:he 
standard  are  quite  common,  and  are  now  being  used  to  sene  extent 
in  this  country.  In  designing  teeth  of  this  kind  care  must  be 
taken  to  make  the  arc  of  action  at  least  as  great  as  the  circula 
pitch,*  otherwise  the  teeth  would  not  be  continuous ly  in  mesh 
and  would  probably  come  together  in  such  a 7; ay  as  to  lock  and 
prevent  further  rotation.  The  standard  angle  of  obliquity  of 
action  adopted  by  manufacturers  of  gear  cutters  iv  slightly 
at  variance  with  the  usual  standard  for  cast  teeth,  being  14° 

28’  4-0",  the  sine  of  which  is  .25. 

The  smallest  involute  gear  of  standard  proportion  that 
will  mesh  correctly  with  a rack  of  thes&me  pitch,  contains 
30  teeth;  however,  this  difficulty  is  remedied  by  slightly  cor- 
recting the  points  of  all  the  teeth  in  the  set,  so  th  t a gear 
of  twelve  teeth  may  mesh  with  any  of  the  other  gears  of  the  same 
pitch.  The  profilea  of  these  teeth  may  be  drawn  with  almost 
exact  accuracy  by  circular  arcs  with  their  centers  on  the  base 
circle,  and  the  values  of  these  radii  for  a 15.  involute  have 
been  carefully  worked  out  by  Mr.  G.  B.  Grant  of  the  Grant  Goer 

Works.  These  values  arc  given  in  the  following  table. 

192 


. i i 1 4 


. 


* ■ 


r 


91 


table  gf  radii  for  15 ° 


INVOLUTE 


! TT 

XI  » 


Centers 


on  Base  Line. 


Divide  by  the  Multiply  by  the 


T*  0 ,ri  -f  Vs 

Diametral  Pitch 

Circular  Pitch. 

Face 

Flank 

Face 

Flank 

Radius 

Radius 

Radius 

Radius 

10 

2.28 

.69 

.73 

.22 

11 

2 .40 

.83 

.76 

.77 

10 

2.51 

;96 

.80 

.31 

13 

2.62 

1.09 

.83 

. 34 

14 

2.72 

1.22 

.87 

.39 

15 

2 .82 

1.34 

.90 

.43 

10 

O CO 

*~J  9 -u 

1.46 

. 9*^ 

.47 

17 

5.02 

1.58 

.96 

.50 

18 

3.12 

1.69 

.99 

.54 

IS 

3.7,7 

1 .79 

1.03 

.57 

20 

3.32 

1.99 

1.06 

.60 

21 

3 .41 

1.98 

1.09 

• c*  rr 

• 0 

22 

3.49 

2.06 

1.11 

4* 

• CO 

O " 

A L/ 

5.71 

2.15 

1.13 

.39 

04 

/! 

• ‘"A" 

2.24 

1.16 

.71 

05 

5.71 

2.33 

1.18 

.74 

03 

3.78 

O * O 

1.20 

.77 

27 

3.85 

2.50 

1.23 

.80 

28 

PT  QO 

-•  # Cj 

2.59 

1.25 

• 87 

29 

3 • ss 

2.69 

1.27 

.05 

30 

4.06 

2.76 

1.29 

.08 

51 

4.13 

2.85 

1.51 

.91 

VO 

4.20 

2.93 

J.  • w — 

* 

• «-*  v> 

O 4) 

/.  O Q 

• .O  *s 

3.01 

“I  *2  A 

J.  • O c 

■ oe 

34 

4.33 

3.09 

"1  rr  r> 

1 » -JO 

.99 

bb 

. .39 

3.16 

i;39 

1.01 

36 

4.45 

3.23 

1.41 

1.03 

37-4-0 

4.20 

1.34 

41-4-5 

4.63 

1.48 

43-51 

5.06 

1.61 

37-60 

5.74 

1.83 

61-70 

6.52 

2.07 

• 

71-90 

7.72 

2.46 

91-120 

9.78 

3.11 

121-180 

13.38 

4.26 

181-360 

21.62 

6.88 

92 


It  will  be  noticed  that  this  table  1 3 for  13°  involute 
end  therefore  does  not  note  the  standard  form  for  cut  gears. 

The  forms  given,  however,  may  be  used  on  the  drawing,  because  in 
cutting  a gear  the  workman  needs  only  to  know  the  number  of  the 
cutter,  and  all  that  is  required  on  a drawing  is  an  approximate 
representation  of  the  tooth  profile.  The  table  also  gives  values 
down  to  a ten  tooth  gear,  while  the  standard  cut  gear  sets  only 
rim  down  to  twelve  teeth.  This  is  theoretically  the  smallest 
standard  involute  gear  that  will  have  an  arc  of  action  equal  to 
the  circular  pitch,  however,  and  in  the  ten  and  eleven  tooth 
gears  the  error  is  so  slight  that  it  is  practically  unnoticeable. 

It  was  found  necessary  to  devise  a separate  means  of 
drafting  the  rack.  The  tooth  is  drawn  in  the  usual  way,  the  sides 
of  the  tooth  making  angles  of  15°  with  the  lines  of  centers  from 
the  root  line  to  a point  midway  between  the  pitch  and  the  addendum 
lines.  The  outer  half  of  the  face  is  formed  by  a circular  arc, 
with  its  center  on  the  pitch  line  and  its  radius  equal  to  2. 10" 
divided  by  the  diametral  pitch  or  0.67"  multiplied  by  the  cir- 
cular pitch.  The  radius  of  the  fillet  at  the  root  of  the  tooth 
is  taken  as  l/4  of  the  widest  part  of  the  tooth  space. 

Standard  Involute  Cutters.  - Brown  and  Sharpe,  the 
leading  manufacturers  of  formed  gear  cutters  in  this  country, 
furnished  involute  cutters  in  sets  of  8 for  each  pitch  as  follows: 

No.  1 will  cut  gears  from  135  teeth  to  a rack 

2 " " " " 55  " " 134  teeth 


- 

Joa  t o Lvp:  i * . f si!  t ,•  f«  " ot  two  I 

<■ 

' ' ' £ • | 


- . 


4 

wi  1 1 

cut 

gears 

from 

26 

teeth 

to 

54 

teeth 

5 

t» 

K 

» 

it 

21 

u 

I! 

25 

IT 

6 

it 

11 

r? 

ft 

17 

it 

n 

20 

11 

7 

ft 

It 

u 

tt 

14 

ti 

tt 

IS 

ft 

8 

ii 

11 

n 

it 

12 

ii 

it 

13 

ft 

When  more  accurate  tooth  forms  are  desired  they  also 
furnish  cutters  to  order  of  the  half  sizes  making  a set  of  15 
cutters  instead  of  8. 

These  cutters  are  commonly  based  on  diametral  pitch 
and  are  made  in  the  follovjlng  sizes:  from  1 to  4 by  quarters: 
from  4 to  6 by  halves:  from  6 to  16  by  whole  numbers;  from 
13  to  52  by  even  numbers  only;  then  36,  58,  40,  44,  48,  50,  56, 
SO,  64,  70,  80,  and  120.  However,  they  also  furnish  cutters  at 
a slightly  greater  cost  based  on  circular  pitch,  and  the  sizes 
vary  as  follows;  from  l/8M  to  1”  by  sixteenths;  from  1"  to  1 l/2" 
by  eighths,  and  from  1 l/2n  to  3n  by  quarters. 

Action  of  Involute  Teeth.  - Fig.  35  illustrates  very 
clearly  the  action  of  a pair  of  involute  teeth.  Let  the  circles 
A and  B represent  the  base  circles  of  a pair  of  involute  gears, 
the  pitch  circles  of  which  would  be  the  circles  described  about 
A and  B with  radii  of  AC  and  BC  respectively.  Imagine  a cord 
attached  to  A at  L extending  around  the  circumference  to  the 
point  D,  from  there  direatly  ©.cross  to  E,  and  abound  the  circum- 
ference of  B to  Me  Let  the  central  point  of  the  string  be  per- 
manently marked  in  some  manner  and  be  denoted  by  C.  Now  rotate 

A in  the  direction  of  the  arrow  and  trace  the  path  of  the  point 

192 


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*m  -m# 


■ 


' . 

■ 


. 


*4  $ K fr 


■ 


' 


' 


C on  the  surface  of  A extended,  on  the  surface  of  B extended,  and 
also  its  actual  path  in  space.  It  is  evident  that  these  throe 
curves  will  be  CG,  OH,  and  CJ,  and  that  CG  and  CH  will  be  por- 
tions of  the  involutes  of  the  two  base  circles  A and  B.  ¥ ow 
reverse  the  rotation  of  B and  rewind  the  string  on  E until  C 
reaches  the  point  K.  During  this  motion  it  will  complete  the 
tooth  forms  OF  and  Cl.  Bearing  in  mind  that  C is  always  the 
point  of  contact  of  the  teeth,  its  path  is  evidently  JE  and 
coincides  exactly  with  the  line  of  pressure  between  the  teeth, 
since  the  line  CD  is  always  normal  to  the  involute  curve  it  is 
generating.  If  the  centers  A and  B should  be  misplaced  slightly 
on  account  of wear  in  the  journals,  a uniform  velocity  ratio 
would  still  be  transmitted  because  the  no male  would  still 
pass  through  the  point  C.  The  only  result  of  this  shifting 
of  centers  would  be  to  change  the  obliquity  of  pressure  of  the 
teeth  and  the  length  of  the  arc  of  contact.  The  outlines  of 
the  teeth  would  not  be  changed  a particle. 

56 . Cycloidal  System.  - The  cycloidal  sy  tw  although 

the  oldest  is  not  so  popular  as  the  involute  system,  an'7  seems 

to  be  gradually  going  out  of  use.  Mr.  Grant  in  his  "Treatise 

or  Gear  Wheels"  says:  "There  is  no  core  need  for  two  different 

kinds  of  tooth  curves  for  gears  of  the  same  pitch  than  there 

is  need  for  different  kinds  of  threads  for  standard  screws,  or 

of  two  different  kinds  cf  coins  of  the  same  value,  and  the  cycloi 

dal  tooth  would  never  be  missed  if  it  were  dropped  altogether. 

But  it  was  the  first  in  the  field,  is  simple  in  theory, 

192 


is 


/ 


~ 


r 


95 


easily  dram,  has  the  recommendation  of  many  Fell  m oaring  teachers, 
and  holds  its  position  by  means  of  human  inertia,*  or  the  natural 
reluctance  of  the  average  human  mind  to  adopt  a change  particu- 
larly a change  for  the  better"  „ This  view  is  probably  a little 
biased,  but  nevertheless  there  is  a great  deal  of  sound  truth  in 
it.  The  proportion  of  machine-cut  cycloidal  teeth  to  machine-cut 
involute  teeth  is  very  small,  but  in  some  classes  of  Fork,  and 
especially  when  the  loads  are  heavy,  they  are  still  used  exten- 
sively . 

Form  of  Tooth.  - The  outline  of  a cycloidal  tooth  is 
made  up  of  two  curves.  The  faces  of  the  teeth  are  epicycloids 
and  the  flanks  are  hypocycloids , with  two  exceptions,  namely, 
internal  gearing  and  racks.  In  the  former  case  the  faces  are 
hypocycloids  and  the  flanks  are  epicycloids,  while  in  the  latter 
both  curves  are  plain  cycloids.  (When  a circle  rolls  on  a fixed 
straight  line  the  path  generated  by  any  assumed,  point  of  the  circle 
is  a cycloid:  should  the  circle  roll  on  the  outside  of  another 
circle,  the  path  of  this  point  would  be  an  epicycloid,  and  should 
roll  on  the  inside  of  another  circle,  it  would  be  a hypo- 
cycloid  . ) 

These  rolling  circles  are  generally  spoken  of  as  de~ 
scribing  circles,  and  their  size  determines  the  for'*'  of  the  tooth, 
the  arc  of  contact,  and  the  angle  of  obliquity  of  action.  The 
angle  of  obliquity  in  the  system  is  constantly  changing,  but  it« 
average  value  when  the  proportions  of  the  teeth  ahe  standard,  is 
about  15°,  the  same  as  in  involute  gearing,  The  circle  upon  ’■’■•h'h 

the  describing  circles  are  rolled  is  the  pitch  circle. 

192 


' 


' 

V. 

: - 

' 

" 


Then  the  diameter  of  the  rolling  circle  is  equal  to 
the  radius  of  the  pitch  circle,  the  flanks  of  the  teeth,  are 
radial,  and  when  it  is  smaller  than  the  radius  of  the  pitch 
circle,  the  flanks  of  the  teeth  are  undercut.  In  addition  to  the 
objection  that  undercut  teeth  are  weak,  the  amount  of  undercut 
must  be  very  slight  if  the  teeth  are  to  be  cut  with  a rotating 
cutter.  Fig.  33  shows  the  limit  of  undercut  possible  with  a ro- 
tating cutter;  the  width  of  the  space  must  either  remain  constant 
or  decrease  as  it  approaches  the  -root  line. 

The  sane  describing  circle  must  always  be  used  for 
these  parts  of  the  teeth  which  work  together,  i.e.  the  faces  of 
a tooth  on  the  one  gear  must  be  formed  by  the  same  describing 
circle  as  the  flanks  of  the  tooth  it  meshes  with.  In  interchange- 
able sets  it  is  desirable  to  use  the  same  size  describing  circle 
for  both  the  faces  and  the  flanks  of  all  the  gears  of  the  same 
pitch,  and  the  size  of  the  describing  circle  which  j-  generally 
accepted  as  standard  is  one  whose  diameter  is  equaljtc  the  radius 
of  a twelve  tooth  gear  of  the  same  pitch.  Here  s,gain,  however,  the 
manufacturers  of  gear  cutters  are  at  variance,  and  use  15  teeth 
as  the  base  of  the  system.  This  does  not  mein  that  the  15  tooth 
gear  is  the  smallest  gear  in  the  set,  but  •imply  means  that  smaller 
gears  will  have  undercut  flanks. 

The  profiles  of  these  teeth,  as  in  the  case  of  involute 

teeth,  may  be  very  accurately  represented  by  circular  arcs,  and 

the  following  table  gives  values  for  these  radii,  with  radial 

distances  from  their  centers  to  the  witch  circle  as  determined 

by  Mr.  Grant.  The  centers  of  faces  lie  inside  of  the  pitch 

192 


* 


• • 


1 

' 


' 

* 

■ ■ t, 

. 

' 


97 


circle,  while  the  centers  of  flanks  lie  outside  of  it. 

The  smallest  gear  in  the  set  is  again  one  having  ten 
teeth,  while  the  smallest  one  for  which  standard  cutters  are 
manufactured  is  one  having  twelve  teeth.  The  form,  given  by  the 
table  is  also  slightly  different  from,  the  form  of  standard  cut- 
ters on  account  of  the  difference  in  describing  circles,  but  os 

in  the  case  of  involute  profiles  nay  be  used  on  the  drawing. 

1C  1 


08 


TABLE  OF  RADII  FOR  CYCLOIDAL  TEETH • 


Number  of 

For 

One  Diametral 

TPni 

i.'  V^yJ 

r One  Jn.  Circular 

Teeth  in 

Pitch.  For  any 

Pitch.  For  any 

other 

the 

Gear 

other  pitch  divide 

pitch  multiply  b 

y 

by  that  pitch. 

that  pitch. 

Faces 

Flanks 

Faces 

TP 

lanks 

E^IcbC  "fc 

i 

I nt ’ v 1 s 

Radr 

Dis  „ ! 

I 

Rad.  I 

->-j  1 s # 

Rad . 

jJiS 

Rad . 

Dis  . i 

10 

10 

1 O Q 

-L  0 0 

.02 

8 c 00 

A . no 

* 7 £* 

.01 

r>  - r~ 

— • / J) 

1.87  j 

1 11 

1 

11  ! 

2.00 

.04  | 

11,05 

n rzr\ 

• O A-; 

.33 

.01 . 

rr  rr  A 

0 ,07 

! 12 

12 

2 o01 

i 

.03  | 

_ 

, 34 

.02 

; 

\i4 

— 

13-14 

2.04 

.07  i 

15.10 

9.45 

n rr 

.00 

• Oy? 

4.00 

15l 

Q 

15-16 

■ 

2.10 

.09  j 

7,86 

7 .46 

.37 

» 0 0 

0 po 

1 

~i  ir  s 

] 

17l 

2 

17-18 

2 . 14 

.11 

1 

6.13 

2.20 

,68 

• 0*4: 

1.95 

7-  1 
.... 

20 

19-21 

2.20 

.13  : 

5 , 12 

1.57 

.70 

. 04 

i.:o 

b 

i 

.30 

25 

22-24 

2.23 

.15 

4.50 

1.13 

.72 

.08 

■”  n j 

* 

l 

27 

25-29 

2.33 

,16 

4.10 

on 

• «-/  sj 

.74 

.05 

1.30 

.89 

I 

ry  r-r 

OO 

50-3  S 

2.40 

.19 

5.80 

.72 

n c 

• ' V_ 

- 

•°? 

1.20 

1 

.-3  j 

f 

Ao 

37-48 

2.48 

.22 

3.52 

.S3 

* ■ 

.07 

1.12 

.80  j 

l 

58 

49-72 

1 

2.60 

I 

.25 

^ cr* 

O 

.54 

.83 

.08 

• 

1.06 

• 

.17  i 

97 

i 

73-14-4 

| 2.85 

,28 

3 . 14 

.44- 

.90 

• 

.09 

1.00 

. 

1 

290 

145-300 

2.92 

• 0 X 

3.  CO 

* 08 

,93 

.10 

.05 

i 0 

. 1,. 

Rack 

| pop 

| Cj  • w 

j 

rzA 

i 2.96 



n£. 

• - 

OA 
© - — 

.11 

0/. 

» ~ - 

192 


' 

. 


. 


* 


' 


Standard  Ov&loidal  Cutters 


Broun  and  Shame  furnish 


sets  of  cycloidal  cutters  based  on  diametral  pitch  only,  and  the 
sizes  vary  as  follows:  from  2 tv  3 by  quarters;  from  5 to  1 by 
by  halves;  from  4 to  10  by  whole  numbers,  and  from  10  to  13  by  t 
even  numbers  only. 

Each  set  consists  of  24  cutters  as  follows: 

Cutter  A cuts  12  teeth  Cutter  M cuts  27-29  teeth 


B 

C 


D " 


15 

14 

15 

E " 16 

P " 17 

G 11  IS 

H 19 

I " 20 


Jf f q n no  ft 

& JL o 


K ” 23-24  " 

L n 25-26  " 


jf  f!  30-73  ” 

0 ,J  34-37  n 

P " 38-42  " 

Q,  " 43-49  " 

R " 50-59  » 

S n 30-74  t: 

rn  T?  r?  r;  nr\  f? 

X t <_.'•*  j . 

U " 100-149 

V n 150-249  ,f 

W ” 250  or  more  teeth 
X racks. 


Action  of  Cycloidal  Teeth.  - Fig.  37  illustrates  the 
action  of  a pair  of  cycloidal  teeth.  Let  the  circles  A and  B 
represent  the  pitch  circles  of  a pair  of  cycloidal  gears,  and 
the  circles  D and  E represent  their  rolling  circles.  Let  C 
be  the  pitch  point,  and  let  C^  and  Ce  be  the  points  on  the  circles 
D and  E which  coincide  with  C when  the  teeth  are  in  the  position 
shown.  How  let  the  centers  of  the  circles  A,  B,  D and  E be  fined 


and  rotate  A in  the  direction  indicated  by  the  arrow. 


Let 


' 


. 


■ 


. 


■ 


I f 


the  contact  at  C be  so  arranged  that  the  circles  B,  D,  and 

E are  driven  with  the  sane  peripheral  speed  as  A.  Then  trace 

the  oath  of  the  point  on  the  surface  of  A extended,  on 

the  surface  of  E extended,  and  also  its  actual  path  in  spao  3 . 

These  paths  will  evidently  b9  respectively  the  hypocycloidal 

flank  of  a tooth  CF,  the  epicycloidal  face  of  the  meshing  tooth 

OH,  and  the  path  of  the  point  of  contact  0 J . Nov:  replace  the 

mexhanism  in  its  original  position,  rotate  A in  the  0">  -osite 

direction  and  trs.ee  the  path  of  the  point  Ca  in  the  sane  '-am? or . 

This  forms  the  curves  CG,  Cl,  and  CK,  and  completes  both  tooth. 

forms  and  the  path  of  contact.  As  the  line  of  pressure  between 

the  teeth  which  of  course  coincides  with  the  common  norms.  1 at 

the  point  of  contact  must  always  pass  through  the  point  C in 

order  to  transmit  a uniform  velocity,  the  angle  of  obliquity 

varies  from  PP'  JCL  to  zero  during  the  arc  of  aorr oach,  and. 

from  zero  to  the  KCM,  which  equals  ->\/~  JCL,  during 

the  angle  of  recess.  In  order  to  show  that  with  this  form  of 

tooth,  the  normal  to  the  tooth  profile  at  the  point  of  contact 

always  passes  through  the  pitch  point  C let  us  observe  Fig.  IS. 

It  is  evident  that  the  generating  point  C0  as  well  as  every 

other  point  in  the  rolling  circle,  is  at  any  given  instant 

rotating  about  the  point  of  contact  C of  the  rolling  circle  ^ith 

the  pitch  circle.  Therefore  on  the  instant  in  question  the 

line  CCg  is  a radius  for  the  point  CQ  and.  is  consequently  normal 

at  that  point  to  the  curve  which  C@  is  generating.  Now  referring 

again  to  Fig.  57,  the  point  at  which  the  rolling  circle  is 

192 


* 

; 


■ ' 

■ 


- 


101 


always  in  contact  with  pitch  circle  is  evidently  the  pitch 
point,  and  therefore  the  common  normal  at  the  point  of  contact 
always  passes  through  it, 

57 » Strength  of  Tooth.  - Having  determine  the  proper 
form  of  a gear  tooth  the  next  step  is  to  determine  its  propor- 
tions for  strength.  Owing  to  the  inaccuracy  of  forming  and 
spacing  the  teeth,  it  is  customary  to  provide  sufficient  strength 
for  transmitting  the  entire  load  by  one  tooth,  rather  than  con- 
sidering the  load  distributed  over  the  whole  number  of  teeth 
in  theoretical  contact. 

The  load  on  a single  tooth,  when  the  gears  are  cast, 
from  w ood  patterns,  is  often  concentrated  at  some  one  point, 
usually  an  outer  corner,  on  account  of  the  draft  on  the  teeth 

and  the  natural  warp  of  the  castings.  The  same  result  is  liable 

1 

to  be  produced  when  the  shaft  is  weak  or  when  the  gears  are  not 
supported  on  a rigid  foundation.  However,  in  the  case  of  well 
supported  machine-moulded  or  cut  gears  the  load  may  be  consider  d 
uniformly  distributed  along  the  tooth.  ?or  the  reasons  just  staged 
the  subject  of  strength  of  teeth  will  be  discussed  under  tvo  heads, 
as  follows:  (a)  Strength  of  cast  teeth:  (b)  Strength  of  cut 
teeth. 

(a)  Cast  Teeth.  - In  deriving  the  formula  for  this 
cla.so  of  gearing,  it  will  be  sufficiently  accurate  to  consider  the 
shape  of  the  tooth  as  rectangular,  and  the  load  as  acting  at  the 
outer  end.  The  load  may  be  concentrated  at  one  corner,  or  it  may 

be  uniformly  distributed  along  the  length  of  the  tooth. 

192 


■ 


■ 


’ 


■ 

- 


■ 


102 


(l)  When  the  load  is  concentrated  at 
as  shown  in  Fig.  39,  we  have  the  following: 

The  bending  moment  due  to  the  load  W 
and  equating  thi3  to  the  moment  of  resistance, 

Wh  cos  *,  = -----  - 

6 sin  cc 

where  S is  the  allowable  stress  of  the  material 
S = 5W  sin  2 
t2~ 

The  stress  S is  maximum  when  2 « is  a 

when  a = 45°,  therefore, 

3W 

max . S = ~ 

t2 


an  outer  corner, 

= Wh*  = Wh  cos 
we  get , 

. From  this 

(74) 


maximum  or 


(75) 


or. 


(2)  When  the  load  is 

the  length  of  the  tooth,  we  have 

moment  and  the  resisting  moment, 

t2fs 


Wh  = 


from  which 


S = 


6Wh 


t2f 


distributed  uniformly  along 
by  equating  the  bending 


(76) 


Equating  the  stresses  given  by  (73)  and  (76),  we  got 

f = 2h  = 1.4p’  (7-) 

when  h = 0.7p*  and  p’  = the  circular  pitch. 

Although  as  shown  by  (77),  the  theoretical  length 

of  face  at  which  the  teeth  will  be  of  equal  strength  for 

both  cases  of  loading  is  1.4p',  a well  known  American  engineer, 

C.  V.  Hunt,  taking  his  data  from  actual  failures  in  his  own  work 

work,  states  that  the  face  should  be  about  2p’  in  order  to 

192 


. 


■ ■ ■ 


v 


. 


. 

" • 


. 


• 

' 


105 


satisfy  this  condition. 

The  seeming  discrepancy  between  theory  and  actual  results 
may  be  easily  explained,  when  one  takes  into  consideration  the 
fact  that  even  though  the  load  may  be  entirely  concentrated  at 
t ■- e corner  at  the  beginning  of  application  of  pressure,  it  is 
very  probable  that  before  the  full  pressure  is  brought  to  bear, 
a slight  deflection  of  the  outer  corner  will  cause  the  load  to 
be  disturbed  along  a sconsiderable  length  of  the  face.  Another 
condition  also  which  adds  to  the  length  of  the  face  is  that  of 
proper  proportions  for  wearing  qualities,  and  in  some  cases  the 
faces  are  made  extra  long  for  that  purpose  alone.  It  is  customary 
in  American  practice  to  make  the  face  2 to  3 times  the  circular 
pitch,  the  length  of  the  face  increasing  as  the  quality  of  the 
work  improves  3 

Common  proportions  for  cast  teeth  may  be  found  in  Kent 
p.  889.  The  values  given  in  columns  two  and  five  are  used  quite 
frequently.  The  thickness  of  the  tooth  is  given  in  column  five 
as  0.475p’:  now  substituting  the  values  for  h and  t given  in 
Kent's  in  (78)  we  get 

W = O.O54  Sp’f  (78) 

This  formula  has  the  same  form  as  the  well  known  Lewis 
formula  derived  below.  It  is  a rather  difficult  matter  to  give 
■ proper  values  for  3 in  (78)  for  different  conditions,  but  the  • 
table  in  Kent  p.  901  or  Fig.  42  may  be  used. 

(b)  Cut  Teeth.  - In  1893  Mr.  Wilford  Lewis  of  Wm.  Sellers 
& Co.  presented  at  a meeting  of  the  Engineers’  Club  of  Philadel- 
phia a very  excellent  method  of  calculating  the  strength  of  gear 
* 192 


■ 


• i 


■ 


‘ 


>• 

" 


• • 


■ 


■ 


. 

• mi  ' 'r  f ' * 


" 


i>  f 


104 


teeth*  His  investigation  was  the  first  one  to  take  into  consi- 
deration the  form,  of  tooth  profile,  and  the  fact  that  the  direction 
of  pressure  is  always  normal  to  this  outline.  It  has  since  that 
time  been  almost  universally  adopted  for  calculating  the  strength 
of  teeth,  when  the  workmanship  is  of  high  grade  as  in  cut  gears. 

In  his  investigation,  Mr.  Lewis  assumed  that  at  the 
beginning  of  contact  the  load  was  concentrated  at  the  end  of  the 
tooth,  with  its  line  of  action  normal  to  the  tooth  profile  in 
the  direction  A B,  Fig.  40.  The  actual  thrust  P was  then  resolved 
at  the  point  B into  two  components,  one  acting  radially  producing 
pure  compression,  and  the  other  W acting  tangent ly,  When  the 
material  of  which  the  gears  are  made  is  stronger  in  compression 
than  in  tension  this  radial  component  adds  to  the  strength  of 
the  tooth,  and  when  the  tensile  and  compressive  strength  are 
approximately  equal,  it  is  a source  of  weakness.  However,  in 
either  case  the  effect  is  slight  not  exceeding  10^,  and  in  the 
original  investigation  was  neglected  altogether. 

The  strength  of  the  tooth  may  now  be  determined  by 
drawing  through  B,  Fig.  40,  a parabola  which  is  tangent  to  the 
toothprofile  at  some  point  D.  This  parabola  then  encloses  a 
cantilever  beam  of  uniform  strength,  the  weakest  section  of 
the  tooth  then  lies  along  the  line  DE,  and  whose  strength  is 
evidently  a measure  of  the  strength  of  the  tooth.  The  problem 
now  is  to  find  an  expression  for  W in  known  terms* 

By  similar  triafigles 
„ t2 

h = (79) 

4x 


192 


' 

■ 


... 

■ 


... 


* 

* 


m 


' 


105 


Equating  the  bending  moment  to  the  moment  of  resistance 
^2  ~ 


Wh  = 


or  W = ^ fSx 

Dividing  and  multiplying  by  p’ 


W = Sp’fy  (80) 

O V 

when  v = .-—h-  is  a factor  depending  upon  the  pitch  and  the  form 
3p' 

of  the  tooth  profile.  Values  of  y and  S are  tabulated  in  Kent, 
p.  901,  also  in  Pig.  41  and  42.  For  cut  gears  f varies  from  2 l/ 
to  4 times  the  circular  pitch.  In  some  cases  when  the  speed  and 
the  conditions  otherwise  are  not  favorable  in  regard  to  wear,  f 
i3  made  even  greater  than  4 times  the  circular  pitch.  For  pro- 
portions of  standard  cut  teeth  see  Kent,  p.  890  and  column  ten 
of  the  table  on  p.  £89. 

58.  Methods  of  strengthening  Teeth.  When  it  is  desir- 
able to  have  the  teeth  of  a gear  extra  strong,  any  one  of  the 
five  following  methods  of  strengthening  may  be  used:  (a)  shroud- 
ing; (b)  using  short  teeth;  (c)  increasing  the  angle  of  obli- 
quity; (d)  stub  tooth;  (e)  using  helical  teeth;  (f)  using  a 
butressed  tooth. 

(a)  Shrouding.  - The  gain  in  strength  due  to  shrouding 

depends  upon  the  face  of  the  gear,  the  effect  being  more  marked 

in  the  case  of  a narrow  face  than  in  a wider  one.  Wilford 

Lewis  considers  shrouding  bad  practice,  However,  when  the  face 

equals  2 l/2  times  the  circular  pitch,  he  has  demonstrated  by 

a crude  theoretical  investigation  that  single  shrouding  (Fig. 

192 


■■ 

. 


■'  . 


. 


. 

. 


■ 


■ 


103 


43, a)  will  increase  the  strength  at  least  10/,  and.  double  at 

least  30/«  Single  shrouding  is  illustrated  by  Fig.  43  (a),  double 

by  Fig.  43  (b)*  and  half  by  Fig.  43  (c)« 

(b)  Short  Tooth,  - Gear  teeth  whose  heights  are  less 

than  that  given  by  common  proportions  are  considerably  stronger, 

and  furthermore,  they  run  with  less  noise.  In  this  country  C.  *7. 

( 

Hunt  advocates  this  type  of  tooth,  and  the  following  proportions 
for  involute  teeth  are  the  ones  he  has  successfully  used  on  gears 
for  coal  ho  i/s  ting  engines  and  similar  machinery. 


Addendum  = 0.2 

p' 

(SI) 

of  ge-ar  = 2p’ 

1 

(82) 

( 

Clea^ance=  .05 

(p’  * 1") 

(83) 

They'  following 

table  used  by  Hunt  gives 

working  and 

maximum  loacws  on  a cast  iron  spur  gear  of  °0  teeth,  which  is  the 
smallest  he  uses. 


i 7- 

'Circular  / 

_ . , t 

Load  in  Pounds 

Circular 

Pitch 

Load  in  P ound 3 

i 

i 

i Pitch  , 

. 

! 

Working  | 

l 

Maximum 

Working 

Maximum 

l / ’ 

i i/' 

i 

1320 

1650 

2 1/4 

6700 

p.  2Q  c 

1 1/4 

2300 

2600 

2 l/2 

8300 

1050Q 

1 1/2 

3000 

3700 

2 3/4 

10000 

12500 

j 1 3/4 

4100 



5000 

3 



12000 

j 14800 

(c)  Increasing  Angle  of  Obliquity.  - The  gain  in  strength 
due  to  increasing  the  angle  of  obliquity  is  shown  in  Fig.  44 
which  figure  consists  of  the  left  half  of  a tooth  of  22  l/%°  ob- 
liquity and  the  right  half  of  a tooth  having  the  same  witch  but 

192 


■ 

. . " 

- 

* 

■ • 

• 

- 


• 

- ' 

. 

• • • « 

. 


• • •;  ' ' 


' 


' 


107 


having  an  angle  of  obliquity  equal  to  15°.  1Toy;  the  factor  y which 

2x 

appears  in  Lewis*  formula  is  equal  to  wp  , from  which  it  is 
evident  that  an  increase  of  x when  p?  remains  constant  will 
result  in  an  increase  of  y and  consequently  an  increase  in  the 
strength  of  the  tooth.  This  increase  of  x im-  shown  in  the 
figure » 

A further  advantage  aside  from  the  increase  of  strength 
lies  in  the  fact  that  the  size  of  the  smallest  pinion  which 
will  mesh  with  a rack  without  correction  for  interference 
diminishes  rapidly  as  the  angle  of  obliquity  increases.  Thus 
with  an  angle  of  obliquity  of  15°  the  30  toothed  pinion  is 
the  smallest  one  which  can  be  used  without  correction,  while 
with  an  obliquity  of  22  l/g°  the  smallest  gear  in  an  uncor- 
rected set  is  theoretically  14,  but  practically  it  may  be  re- 
duced to  12  * 

(d)  Stub  Tooth.  - Another  method  of  strengthening 
gear  teeth,  which  is  now  being  introduced  quite  extensively  in 
automobile  transmission  gears  consists  of  a combination  of  (b) 
and  (c),  and  is  known  as  stub  tooth.  The  angle  of  obliquity 
used  is  20o,  and  the  following  table  gives  tooth  dimensions  as 
recommended  by  the  "Fellows  Gear  Shaper  Co."  of  Springfield, 
Vermont . 

i oo 
jl 


■ 


' i 


- 

i • ini..** 

' 


107 


TABLE  OF  STUB  TOOTH 



DIMENSIONS  1 

Pitch 

Thickness  on 
Pitch  Line 

Addendum 

Dedendum 

4/5 

e 3925 

.200 

.250 

' 

5/7 

.SIS 

. 142  S 

.1785 

3/8 

.2617 

.125 

1 

1 r.,-2  0 j 

1 

7/9 

.2243 

olll 

1 

.1389 

8/10 

.1962 

.100 

.125 

jo/n 

. 1744 

. 1909 

.1137 

10/12 

.157 

uOSoo 

,1042 

112/14 

! 

.1308 

0.0714 

0007 

a \j  ■ ^ o 

(e)  Helical  Teeth.  - Gears  having  accurately  made 
helical  teeth  and  if  properly  supported,  will  run  much  smoother 
than  ordinary  gears.  In  the  latter  form  of  gearing  there  is  a 
time  in  each  contact  when  the  whole  load  is  concentrated  on  the 
edge  of  the  tooth,  thus  having  a leverage  equal  to  the  height 

of  the  tooth.  With  helical  gearing,  however,  the  points  of 
contact  at  any  instant  are  distributed  over  the  entire  working 
surface  of  the  tooth  or  such  parts  of  two  teeth  in  contact  at  the 
same  time.  Therefore  the  mean  lever  arm  with  which  the  load 
may  act  in  order  to  break  the  tooth  cannot  be  more  than  half 
the  weight  of  the  tooth.  Prom  the  above  remarks  it  follows  that 
the  helical  teeth  are  considerably  stronger  than  the  straight  ones, 

(f)  Buttressed  Tooth.  - The  buttress  or  ho ok- tooth 

gear  can  be  used  in  cases  where  the  power  is  always  transmitted 

in  the  same  direction.  The  loaded  side  of  the  tooth  has  the 

1S2 


. 

. 

••  ■ 


. 


' U ■:  k ,1  , ■ . 


IOC 


usual  standard  profile,  while  the  back  side  has  a greater  obliquity 
as  shown  in  Fig.  45.  To  compare  its  strength  to  that  of  the 
standard  tooth,  use  the  following  method:  make  a drawing  of  the 
two  teeth  and  measure  their  thickness  at  the  top  of  fillet;  then 
the  strength  of  the  hook  tooth  is  to  the  standard  as  the  square 
of  its  thickness  is  to  the  square  of  the  thickness  of  t’-'e  stan- 
dard tooth. 


no 
^ «_/ 

the  arm  of  a 
flexure,  and 
is  uniformly 
Letting 


then 


Gear  Wheel  Proportions.  - Arms.  - In  proportioning 
gear  it  may  be  considered  a cantilever  beam  under 
the  usual  assumption  regarding  the  load  ic  that  it 
distributed  over  the  arm. 

— = section  modulus  of  the  arm  at  the  center. 


c 

S = allowable  fiber  stress. 

T = twisting  moment  transmitted,  by  the  gear 

n = number  of  arms 
T _ SI 


(94) 


n c 

From  (84)  a value  of  — may  be  found  from  which  the 

0 

dimensions  of  the  adopted  sections  may  readily  be  determined. 

Fig.  46  shows  four  types  of  arms  used  in  gear  construction,  of 
which  (a)  (b)  and  (d)  are  intended  for  large  gears  and  (c)  for 
lighter  gears,  though  quite  frequently  it  is  used  for  heavy  gears. 
The  method  of  laying  out  is  plainly  shown  in  Fig.  46  (e).  Assuming 
the  proportions  as  shown  in  (e),  (84)  reduces  to 

h = AjZP-l  ( 0" ) 


nS 


192 


‘ 


110 


For  web  centers  make  the  thickness  of  the  web  l/?T  p*. 

Rimo*  - Calculations  for  rim  dimer* 1 ions  are  of  little 
value,  and  in  actual  designing  empirical  formula,  give  the  best 
results.  Fig.  46  shows  the  proportions  used  for  the  various 
rim  sections  commonly  met  with  in  gear  construction. 

Hubs.  ~ The  common  solid  hub  should  have  a reinforce- 
ment of  metal  over  the  key  as  shown  in  Fig.  4-6.  In  olace  of  a 
solid,  hub  one  that  is  split  may  be  used,  thereby  reducing  the  cool 
ing  strains*  in  the  wheel  and  at  the  same  time  permitting  an  easy 
adjustment  on  the  shaft.  Keys  should  always  be  placed  under  an 
arm  in  the  case  of  a solid  hub,  and  in  a split  hub  amrom.imately 
at  right  angles  to  the  center  split. 

The  following  formulas  by  Herman  Johnson  published  in 
the  American  Machinist,  Jan.  14,  1904,  are  applicable  to  gear 
wheels,  pulleys  and  other  machine  parts.  They  represent  the 
actual  practice  of  four  large  manufacturing  concerns. 


Diameter  of  Hub  in  Terms  of  the  Bore. 

Mature  of  Power  Transmitted.  Cast  Iron  Steel  Castings. 

Heavy,  very  great  shock  2d  1 3/4  »•  l/sT! 

1 3/4  + i/8  " 1 5/3  + 3/le" 

1 5/8  1/8"  1 1/2  d + 1/4" 


Standard,  medium  shock 
Light,  no  shook 


19? 


' 


' 


Ill 


Length  of 

Hub  in  Terms  of 

the  Bore. 

Bearings  3d 

to  4d 

Levers 

1 1/2  d 

Gear  Wheels  1 

5/4d  to  2 l/4d 

Pulleys 

2/3  Pace 

Hand  Wheels 

1 1/5?  d to  2 d 

Truck  Wheels 

2d  to  ? l/4d 

60.  Definitions,,  - (l)  A spur  gear  is  a toothed  gear 
the  teeth  of  which  are  parallel  to  its  axis. 

(?)  When  a large  and  a small  gear  mesh  together  the 
large  one  is  called  the  gear  and  the  small  one  the  minion, 

(3)  A rack  is  a gear  wheel  with  an  infinite  radius, 
or,  in  other  words,  a straight  bar  with  gear  teeth  formed  on  it. 

(<_•)  An  internal  gear  is  one  having  its  teeth  on  the 
inside  of  the  rim  and  is  also  called  an  annular  gear n 

(5)  A gear  blank  is  the  solid  wheel,  from  which  the 
gear  is  formed  before  the  teeth  are  cut  e 

(c)  The  face  of  a gear  is  the  wiCth  of  the  gear,  i.e. 
the  length  of  the  tooth. 

(7)  The  pitch  circles  of  a pair  of  gears  are  imaginary 
circles,  the  diameters  of  which  are  the  same  as  the  diameters  of 
a pair  of  friction  gears  that  would  replace  the  spur  gears. 

(8)  The  base  circle  is  an  imaginary  circle  used  in 
involute  gearing  to  generate  the  involutes  which  form  the  tooth- 
profiles  . 

(?)  The  describing  circle  i ' an  imaginary  circle  use'1 
in  cycloidal  gearing  to  generate  the  epicycloidal  and  hypo cy- 
cloidal curves  which  form  the  tooth  profiles.  There  are  two  de- 
scribing circles  used,  one  inside  and  ere  outside  of  the  -itch 

192 





' 


' 


■ 


.... 

■ 


■ , ;.iij 


. 


. 

. 

. 

112 


circle,  and  are  usually  the  sane  size. 

(10)  Angle  of  obliquity  of  action  is  the  inclination 

of  the  pressure  between  a pair  of  mating  teeth  to  a line  drawn  tan- 
gent to  the  pitch  circle  at  the  pitch  point,  i.e,  the  angle  DCF  in 

Fig.  47. 

(11)  Arc  of  approach  is  the  arc  measured  on  the  ^itch 

circle  of  3,  gear  from  the  position  of  the  tooth  at  the  beginning 

of  the  contact  to  the  central  position,  i.e.  the  arc  EC  in  Fig.  47. 

(12)  Arc  of  recess  is  the  arc  measured  on  the  pitch 

circle  from  the  central  position  of  the  tooth  to  its  position  where 
contact  ends,  i.e.  the  arc  Cl  in  Fig.  47, 

(13)  Arc  of  action  is  the  sum  of  the  arcs  cf  approach 
and  recess, 

(14)  Diametral  pitch  is  the  number  of  teeth  divided  by 
the  pitch  diameter.  It  is  not  any  diameter  on  the  gear,  but  is 
simply  a convenient  ratio  to  use, 

(15)  Circular  pitch  is  the  circumference  of  the  pitch 
circle  divided  by  the  number  of  teeth,  and  is  equal  to  the  distance 
from  one  tooth  to  a corresponding  point  on  the  next,  measured  on 
the  pitch  circle. 

(16)  Chordal  pitch  is  the  distance  from,  one  tooth  to  a 

corresponding  point  on  the  next  measured  on  a chord  of  the  pitch 
circle  instead  of  the  circumf erenae . This  is  only  used  for  "'aking 
the  drawing,  or  if  the  teeth  are  t.to  be  formed  on  a wood  pattern, 
by  the  pattern  maker,  ’ 

(17)  Thickness  of  the  too(th  always  means  the  thickness 

l 

i9te 


on  the  pitch  line 


• ' 

■ . 


■ 


. 


■ 

• 

■ 

• ■ 

TO' 


113 


(18)  Tooth  space  means  the  width  of  the  rmace  on  the 

pitch  line. 

(IS)  Backlash  is  the  difference  be  twee  - tooth  srr-c- 

and  thickness  of  tooth. 

(80)  Clearance  is  the  difference  between  addendum  and 
dedendum,  or,  in  other  words, ’the  amount  of  space  between  the  root 
of  a tooth  and  the  point  of  the  tooth  which  meshes  with  it. 

(21 ) Addendum  is  the  radial  distance  from  the  witch 
circle  to  the  ends  of  the  teeth. 

(22)  Dedendum  is  the  radial,  distance  from  the  pitch 
circle  to  the  roots  of  the  teeth. 

(23)  The  face  of  a tooth  is  that  portion  of  its  profile 
which  lies  outside  of  the  pitch  circle. 

(24)  The  flank  of  a tooth  is  that  portion  of  the  Pro- 
file which  lies  inside  the  pitch  circle. 

(25)  Line  of  centers  is  the  line  pa'"  sinr  £ erectly 
through  both  centers  of  a pair  of  gears. 

(26)  Pitch  point  is  the  point  at  which  the  pitch  circles 
of  the  tvro  gears  are  tangent  to  each  other. 

(27)  Velocity  ratio  always  ^eans  t]  o ratio  of  "we  revo- 
lutions of  the  driver  to  the  revoltuions  of  t’  o driven. 

ttepftrtant  • Points , - (l)  The  word  w diavoterf:  when  used 
in  connecton  with  gears  is  always  understood  to  ween  witch  dia- 
meter. 

(2)  Circular  pitch  is  an  actual  dimension  and  is  always 
in  inches;  the  diametral  pitch,  however,  is  only  a ratio. 

(3)  The  circumference  of  the  pitch  circle  cf  a gear 

192 


114 


must  always  be  a multiple  of  the  circular  pitch  or  number  of 
teeth  in  the  gear  will  not  co”-e  out  even. 

(4)  The  relation  between  diam.etra1  a.v.A  circular  pitch 
is  expressed  by  the  formula  pn 1 = Tf 

(5)  The  relation  between  the  diametral  pitch,  pitch  dia- 

F 

meter  &,nd  number  of  teeth  is  p = — t . 

(S)  For  Brown  and  Sharpe  proportions  the  relation  be- 
tween diametral  pitch,  outside  diameter,  and  number  of  teeth  is 

N + 3 

p = — _ — 

(7)  In  designing  out  gears  care  should  always  be  taken 
to  malie  the  diametral  pitch  3ome  regular  standard  pitch.  Although 
gear  cutters  can  be  obtained,  which  are  haded  on  circular  pitch, 
the-;  are  not  usually  kept  in  stock,  and  gears  designed  to  bo  cut 
with  then  will  almost  always  cause  vexatious  delays  in  the  shop 0 

(S)  In  designing  cast  gears  the  proportion  of  the  teeth 
are  usual  1;^  given  in  terms  of  the  circular  pitch,  and  if  the  teeth 
are  to  be  formed  by  the  pattern  maker,  it  is  just  as  well  to  ignore 
the  diametral  pitch  altogether.  However,  if  the  pattern  is  to  be 
of  metal  its  teeth  will  probably  be  cut  and  they  should  be  designed 
with  that  end  in  view. 

When  two  mating  gears  of  such  size  that  the  circum- 
ference of  one  is  an  even  multiple  of  the  circumference  of  the 
other,  it  is  evident  that  any  tooth  on  the  large  one  will  always 
mesh  with  the  same  tooth  on  the  small  one.  This  condition  ayr 
vat os  inequalities  of  wear,  and  when  the  distances  between  costers 
and  the  velocity  ratio  will  permit  of  a slight  variation,  an  extra 

or  hunting  tooth  is  sometimes  added  to  one  of  the  gears. 

192 


11" 


CHAPTER  IX. 

Bevel  Rearing r 

When  two  shafts  which  intersect  each  other  are  to  be 
connected  by  gearing,  the  result  is  a po.ir  cf  bevel  gears.  The 
origin  of  the  name  is  obvious  as  the  faces  of  the  gears  must  be 
beveled  toward  the  axis  m order  to  fit  each  other.  Occasionally, 
however,  the  shafts  are  inclined  at  an  angle  to  each  other,  but 
do  not  intersect,  in  which  case  the  gears  have  teeth  which  are 
not  in  the  same  plane  with  the  axis,  and  are  called  shew  bevels « 

Friction  bevels  are  quite  common,  and  what  has  been 

said  in  the  previous  chapter  regarding  spur  frictions  will  apply 
in  general  to  bevels  also. 

The  form,  of  tooth  which  is  almost  universally  used  for 
tooth  bevel  gears  is  the  involute.  This  is  probably  due  to  the 
fact  that  slight  errors  in  its  form  are  not  nearly  so  disastrous 
to  the  running  of  the  gears  as  when  the  tooth  curves  are  cycloidal. 

61 „ Processes  of  Manufacture.  - These  gears  may  be 
either  cast  or  cut,  and  are  made  in  both  ways , The  process  of  cast- 
ing is  not  materially  different  from  that  used  in  spur  gearing, 
but  the  process  of  cutting  is  much  more  difficult  on  account  of 
the  continuously  changing  form  and  size  of  the  tooth  from  one  end 
to  the  other. 

As  in  the  case  of  spur  gearing,  there  are  several  dif- 
ferent methods  of  cutting  the  teeth,  some  of  which  form  the  teeth 

with  theoretical  accuracy,  while  others  produce  only  approximately 

192 


. 


. 

. 

' ■ n.  i 


■ 


116 


correct  forms.  Throe  of  these  methods  give  very  accurate  results, 
hut  they  require  expensive  special  machines  and  are  used  only 
when  very  high  grade  work  is  desired.  The  three  methods  are:  the 
templet-planing  process  represented  by  the  Gleason  gear  planer: 
the  temp let -grinding  process,  represented  by  a machine  manufactured 
by  the  Leland  and  Paulcover  Company  and  the  moulding-planing  pro- 
cess , represented  by  the  Bilgram  bevel  gear  planer 0 

In  all  these  processes  the  path  of  the  cutting  tool 
always  passes  through  the  apex  of  the  cone,  i.e«  the  point  of 
intersection,  of  the  two  shafts,  and  consequently  the  grower  con- 
vergence is  given  to  the  tooth.  With  a formed  rotating  cutter, 
however,  it  is  impossible  to  produce  the  groper  convergence  and  in 
many  cases  the  teeth  have  to  be  filed  after  they  are  cut  before 
they  will  mesh  properly.  Nevertheless,  the  milling  machine  is 
very  commonly  used  for  cutting  bevel  gears,  for  the  simple  reason 
that  the  equipment  of  most  shops  includes  a milling  machine, 
while  comparatively  few  do  enough  bevel  gear  cutting  to  justify 
the  purchase  of  an  expensive  special  machine  for  that  purpose, 

62,  Form  of  Teeth,  - When  the  gears  are  plain  frictions 

\ 

it  is  evident  that  the  face  of  the  gears  must  be  frustrums  of  a 
pair  of  cones  whose  vertices  are  at  the  point  of  intersection  of 
the  axes.  These  cones  may  now  he  considered  the  pitch  cones  of  a 
pair  of  tooth  gears,  and  the  teeth  may  be  generated  in  a manner 
analagous  to  the  methods  used  for  spur  gearing.  In  discussing 
the  method  of  forming  the  teeth,  the  involute  system  only  wil"1  be 

considered,  as  the  cycloidal  form  is  very  little  used. 

192 


4 


■ . ' 


c 


. 


117 


In  Fig,  48  (this  discussion  refer:'-  to  the  dotted  lines 
only),  let  the  cone  OHI  represent  the  base  cone  of  a bevel  gear 
from  which  the  involute  tooth  surfaces  are  to  be  wrapped,  In  order 
to  simplify  the  conception  of  the  process  of  unwrapping,  imagine 
the  cone  to  be  enclosed  in  a very  thin  flexible  covering,  vdiich  is 
cut  along  the  line  OE0  Now  unwrap  the  covering,  tabing  care  to  keep 
it  perfectly  tight,  and  the  surface  generated  by  the  edge  or  ele- 
ment OF  is  the  desired  involute  surface.  The  -point  E,  while  it  evi- 
dently generates  an  involute  r.  of  the  circle  HI,  is  also  constrained 
to  remain  at  a constant  distance  from  0 equal  to  GE,  or  in  other 
words,  it  travels  on  the  surface  of  a sphere  HAI , From  that  re.  son 
the  curve  FE  is  called  a spherical  involute.  Fig.  49  illustrates 
the  method  of  finding  the  base  cone.  In  this  figure  the  cones  GC1 
and  OFG  represent  the  pitch  cones  of  a pair  of  bevel  gears  whose 
axes  intersect  at  the  point  0 and  whose  element  of  contact  is  the 
line  OE,  The  axis  of  the  cone  OFG  happens  by  mere  accident  to 
coincide  with  the  extreme  right  hand  edge  of  the  cone  OCD0  Now  on 
account  of  the  fact  that  the  ends  of  the  teeth  as  generated  above, 
lie  on  the  surface  of  the  sphere  £EPS , let  us  consider  the  surfaces 
contained  Inside  the  sphere  only.  Pass  a plane  LM  tangent  to  the 

cone  OOD/  at  the  element  OJ,  Then  pass  a second  piano  HP  through 
/ 

the  el/ement  OJ  making  an  angle  with  the  plane  LSI  equal  to  the 

J 

angLjfV  of  obliquity,  and  construct  a cone  OHI  tangent  to  the  plane  W o 
Thir.  cone  is  the  base  cone  for  the  gear,  and  it  evidently  corres- 
ponds exactly  to  the  base  cylinder  of  a spur  gear.  It  is  pi  inly 

^-een  that  as  the  point  of  intersection  of  the  axe  - or  apex  of  the 

192 


' ' ■ 

; • 

; ’ 


• 

V 

. 

■ 


'• 


' 


lie 


pitch  cones  move  farther  away  the  gears  approach  the  spn.rv.goar 
form,  an"  actually  become  spur  gears  when  the  apex  recedes  to  in- 
finity. The  spherical  surfaces  which  should  theoretically  form 
the  tooth  profile  is  a very  hard  surface  to  deal  with  in  pr<?c- 
tice  on  account  of  its  undevelopable  character , arc1  as  is  shown 
in  Fig,  46,  no  appreciable  error  is  introduced  if  the  conical 
surface  CBD  is  substituted  for  the  spherical  surface  Oh D»  This 
cone  CED,  which  is  called  the  back  cone,  is  tangent  to  the  cohere 
at  the  circle  Of,  and  the  pitch  practically  coincides  with  the 
sphere  for  the  short  distance  necessary  to  include  the  entire 
tooth  profile,  Uhen  it  is  desired  to  obtain  the  for-'  of  the 
teeth,  as  is  necessary  in  case  a wood  pattern  or  a formed  cutter 
is  to  be  made,  the  back  cone  is  developed  in  a plane  surface  as 
shown  in  Fig,  i-50,  3Fow  it  is  evident  that  the  surface  which  con- 
tains the  tooth  profile  has  a radius  of  curvature  equal  to  ED, 
so  the  profile  must  be  laid  off  on  a circle  of  that  radius  in 
precisely  the  same  manner  as  it  was  done  in  spur  gearing.  How- 
ever, this  profile  is  correct  for  one  point  only,  namely,  at  the 
large  end.  In  order  to  determine  the  form  of  the  tooth  for  its 
entire  length,  it  is  necessary  to  have  the  profile  of  the  tooth 
at  both  ends.  This  may  be  done  by  developing  the  bac]:  cone  AIJ 
and  proceeding  as  before.  If  a wood  pattern  is  to  be  made, 
templets  are  formed  tof  the  exact  profile  of  the  tooth  at  the 
large  and  small  ends.  These  templets  are  then  wrapped  around 
the  gear  blank  and  the  material  carved  out  to  the  shape  of  J 
templets  o 


192 


' 

i 


. < 


• . ■ : 


119 


65c  Strength  of  Tooth.,  - As  in  the  case  oh  spur  gears, 
formulas  for  the  strength  of  "bevel  gears  will  "be  desired  for  the 
following  two  cases:  (a)  7/hen  the  teeth  are  cast;  (b)  when  the 

teeth  are  cut. 

(a)  Oast  Teeth*  - It  is  sufficiently  accurate  to  consi- 
der the  tooth  as  a cantilever  beam,  the  cross  sections  of  which 
are  rectangular,  and  converge  towards  the  apex  of  he  pitch  cone. 
Furthermore,  the  load  to  be  transmitted  is  assumed  as  acting  tan- 
gentially at  the  tip  of  the  tooth* 

The  following  assumption  with  regard  to  the  distribution 
of  the  pressure  on  the  tooth  seems  reasonable.  Let  the  pressure 
be  so  distributed  that  the  fiber  stress  at  all  points  along  the 
line  of  the  weakest  section  is  equal.  The  reason  for  making  this 
assumption  may  be  explained  as  follows:  assume  two  properly  con- 
structed bevel  , gears  in  mesh,  and  consider  one  of  them  locked 
while  the  other  is  acted  upon  by  a turning  form.  Due  to  the  elas- 
ticity of  the  material,  the  tendency  of  this  force  is;’to  cause  the 
tooth  to  deflect  a small  amount  and  the  deflection  of  any  point 
cn  the  line  of  contact  AB,  Fig.  51  is  proportional  to  its  distance 
from  the  apex  of  the  pitch  oun^c 

The  dimensions  of  the  successive  cross  sections  of  the 
tooth  arc  also  proportional  to  the  distance  from  the  apex.  There- 
fore from  the  known  fact  that  the  deflections  are  proportional :•  to 
the  loads  in  any  series  of  cantilever  beams  of  the  same  breadth 
and  whose  ratio  of  length  to  depth  is  constant,  the  load  at  any 

cross-section  must  likewise  be  proportional  to  its  distance 

192 


' 





1 


• . 


180 


from  the  apex?  Row  since  at  any  point  on  the  line  of  contact.' 

the  load  and  dimensions  of  the  section  are  proportional  to  the 

► 

distance  of  that  point-  from  the  apex,  it  follows  that  the  stress 
along  the  line  of  weakest  section  is  the  same  at  all.  points. 

The  formula  for  strength  of  cast  teeth  may  now  be 
derived  as  follows;  (see  Fig,  51 )« 

By  equating  the  bending  moment  to  the  moment  of  resis- 


tance and  solving  for  dW  >•  we  have 

tSsdl 


t = 


dW  = 


6 h 


h- 


From  the  geometry  of  the  figure,  h=  _L 

J 1 


it- 


and 


(8c) 


Substituting  these  values  in  (86) 

2 / 


dW  = 


St j jdl 

6hr1 


3? ow  the  moment  of  the  force  dY.'  equals  /dW,  or 


dM  = St !2 * * S/SdJ 


Integrating 

M 


St 


6hxTl 


2 


IP  h^l 


2 , ° 
= Stl  >(1 

16h1D1? 


/ - i - 

" 1 ->2 


v - v 


•'87  > 

V ^ ' i 


(88) 


M represents  the' total  turning  moment  around  the 


apex  of  the  pitch  cone,  therefore,  to  find,  the  force  acting 

192 


• 

• 

• 


121 

at  any  point  as  at  the  greatest  addendum,  divide  M by  the  dis- 
tance of  the  point  from  apex.  Let  W*  represent  a force  which 
if  applied  at  the  large  end  of  the  tooth  would  produce  a turning 
moment  equivalent  to  M,  then  ^ 


8- 


, _ M Sti^li 


W'  = 


1- 


18hj 

F 


*7  *7 

V V' 


D- 


(39) 


F Do 

From  Fig.  52  ~ = 1 - _£  substituting  the  value 

h Di 


of  1]_  from  this  equation  in  (89) 


, St-,2f 

W’  = ± 

18h, 


V - ®83 


O \ 


D1<=(D1  -D2; 


(9.0) 


Assuming  the  same  proportions  for  the  teeth  as  in 
spur  gears  (h^  = 0.7  p’  and  t^  = 0.475p’)  and  substituting  these 
values  in  (90) 

(Do)' 


w = .oiesp'f 


From  Fig.  52 


1 


1 + 


D, 


D- 


D- 


(91) 


D, 


1 - f 

X1 


: substituting  this  in 


(91)  we  have  W*  = Sp’frc  1 


in  which  rn  = 0.018 


3-3 


f 


(f)S 


In  good  practice 


£ should  never  exceed  l/c . For  values  of  S for  various  mater- 


ials consult  Fig.  42,  and  the  following  table,  gives  values  of 

f 

m for  different  values  of  — • 


192 


, 


•• 


■ 


' ■ 


.... 


■ 


122 


F 

7i 

cl 

.2 

• £ 

.4 

. 5 

.6 

.7 

.8 

.9 

o 

• 

i — 1 

n 

.049 

.044 

.039 

.035 

n*c 

c 028 

.025 

<•028 

.02 

.018 

(b)  Out  Teeth.  - The  formula  generally  adopted  by  de- 
signers for  calculating  the  strength  of  cut  bevel  teeth,  is  the 
cne  proposed  by  Wilfred  Lewis  in  a paper  before  the  Engineers' 
Club  of  Philadelphia  in  1893.  For  the  assumption  made  in  his 
investigation  consult  Art.  57  (b).  The  formula  as  derived  by 

i 

Hr.  Lefais  is  given  in  Kent  p.  902,  or  may  be  found  as  follows: 

(see  Pig.  53).  In  this  proof  the  same  assumption  regarding  the 

distribution  of  the  pressure  on  the  teeth  will  be  made  as  in  (a) 

above.  Furthermore,  equations  (86)  to  (90)  inclusive  hold  in  the 

present  case.  From  (79)  x = -1-  , so  substituting  this  value 

4h  , 

in  (90)  and  multiplying  through  by  p we  get 

Sp’f 


W?  = 


r”  _ 1 ^ r — 

_ 5 P _ 3 

Dl  - 1)2 

2z 

Dp2  (Dl  -Dp) 

i 

3p  5 

- J 

(S3) 


2x 


Letting  y = pp  t as  in  Art.  57  (b) 


V-  1 _ O y,  T 


= Sp’fyn 

i f 

where  n = 1 - — + 

f 1 


1 


I£LL 


f 


J 


(£:■!) 


, values  of  which  for  various 


ratios  of  -j  are  given  in  the  following  table: 

J i 


. f 

7i 

,1 

.2 

rr 

» o 

.4 

. 5 

. 6 

.7 

.8 

— 

.9 

i i 

o 

n 

.903 

00 

i — 1 

.75 

.655 

.583 

• A. 

.465 

.415 

.57 

^rr  r- 

«*  ».AO  f ’J 

192 


. 


• . 


. 


- 


■ 

■ • ■ 


■ 


As  stated  before 


12? 

should  never  exceed  l/o  and  t^e 
value  cf  f which  is  commonly  used  is  about  2 1/2  p*.  For  values  of’ 
S for  different  materials  at  various  speeds  see  Fig.  42.  In 
obtaining  a value  for  y from  Fig.  41  use  the  formative  number  of 
teeth. 


In  order  to  save  time  the  following  method  may  be  used 
in  finding  the  strength  cf  bevel  gear  teeth: 

(a)  Multiply  the  number  of  teeth  in  the  bevel  gear  by 

^1  (see  Fig.  52)  which  gives  the  formative  number  of  teeth. 

r 

(b)  Find  the  strength  of  a spur  gear  having  this  num- 
ber of  teeth  and  of  the  same  face  and  pitch  as  the  bevel  gear. 

(c)  Multiply  the  strength  of  this  spur  gear  by  the 

f 

value  r taken  from  the  table  for  assumed  value  of  — r . 

11 

64.  Gear  Y/heel  Proportions.  - Arms , - In  bevel  gears 

the  T arm  is  remarkably  well  adapted  for  resisting  the  strains 

that  cone  upon  it  and  for  that  reason  is  quite  extensively 

used  in  gear  wheels  of  large  size.  In  small  gears,  however, 

the  extra  expense  of  construction  more  than  offsets  the  saving 

of  material,  therefore  the  web  and  solid  centers  are  in  common 

use.  Fig.  54  shows  a T arm.  as  applied  to  bevel  gears. 

The  rib  or  feather  at  the  back  of  the  arm  is  added  to 

give  lateral  stiffness,  and  in  the  derivation  of  the  formula 

•for  its  strength  their  effect  will  be  neglected,  since  they  add 

very  little  to  the  resistance  of  the  arms  to  bonding  in  the 

plane  of  the  wheel.  As  in  the  case  of  spur  gears  the  arm  is 

treated  as  a cantilever  beam  under  flexure  and  that  each  arm 

will  carry  — part  of  the  load  transmitted,  by  the  gear. 
n ’ 192 


■ 


■ ■ 

. 

■ 


• 

V 

■ 

? ■ 


124 


Let  b _ thickness  of  the  arm 

h - width  of  arm  at  the  center 
D]_=  pitch  diameter  at  large  end  of  tooth 
S = allowable  fiber  stress 
W ' =equivalent  load  at  large  end  of  tooth 
n = number  of  arms 


W’D 

then  

2n 


Sbh2 

e~ 


from  which 


h = 


35{ 1 D . 


bSn 


1 


f or 


Exercise.  - Assuming  b = 7T  p ’ and  S = C00'n  for  cast 
iron,  establish  the  formula  for  h. 

Rim  and  Feathers  . - Fig.  54  gives  the  proportions  of 
the  rim  and  feathers  commonly  used. 

Hub.  ~ For  hub  dimensions  consult  the  table  given  in 


'i 


Art . 59 „ 


65.  Definitions . - (l)  The  back  cone  radius  is  the 
length  of  an  element  of  the  back  cone.  See  Fig.  50. 

(2)  The  formative  number  of  teeth  is  the  number  of 
teeth,  of  the  given  pitch  which  would  be  contained  in  a complete 
spur  gear  of  a radius  equal  to  the  back  cone  radius. 

(5)  The  edge  angle  is  the  angle  between  o plane  which 
is  tangent  to  the  back  cone,  and  the  plane  containing  the  pitch 
circle.  See  Fig.  50. 

(4)  The  center  angle  is  the  angle  between  a plane, 

tangent  to  the  pitch  cone  and  the  axis  of  the  gear.  See  Fig.  50. 

192. 


. 


. 


. 

. 


. 


_ 


1 or 

(5)  The  out t. lnr~,  angle  ip  the  angle  between  a plant 
tangent  to  the  root  cone,  and  the  axis  of  the  gear.  See  Fig.  f 0 , 
{ 3 ) Backing  is  the  distance  from  the  addendum  at  the 
large  end  of  the  teeth  to  the  end  of  the  hub  which  is  on  the 
large  end  of  the  gear,  measured  parallel  to  the  axis  of  the 
gear.  See  Fig.  50. 

(7)  A miter  gear  is  a bevel  gear  whose  center  angle  is 
4-5°,  and  a pair  of  miter  gears  arc  always  exact  duplicates  of 
one  another . 

Important  points.  - (a)  As  bevel  gear  teeth  taper 
from  end  to  end  they  actually  have  a number  of  pitches  and 
pitch  diameters,  but  in  speaking  of  the  pitch  or  the  diameter, 
the  values  at  the  large  end  are  always  meant. 

(b)  The  formulas  given  in  Kent  p.  POO  for  spur  gears 
may  all  be  used  for  bevel  gears  except  the  ones  in  which  b the 
outside  diameter  appears.  This  diameter  for  a bevel  gear  may 

be  calculated  by  the  following  formula; 

N -4-  2 cos  ...  . 

D = (96) 

P 

when  u is  the  center  angle.  (Derive  this  formula.) 

(c)  The  back  cone  radius  is  used  for  determining  the 
formative  number  of  teeth. 

(d)  The  formative  number  of  teeth  is  used  for  all  pur- 
poses which  pertain  to  the  shape  of  the  teeth;  e.g.  in  selecting 
the  proper  cutter  for  cutting  the  gear,  and  in  obtaining  the 

factor  y for  calculating  the  strength. 

197 


' 


' 

. 


. 


in  diameter  breaks  under  a tensile 


1.  A round  red  2n 
lead  of  190000  pounds.  Find  the  unit  breaking  stress. 

2 . Using  same  unit  stress  as  in  problem  ( 1 } That  load  will 
cause  a 1 l/4"  square  red  to  fail  by  pulling  apart? 

c.  Assuming  a working  stress  equal  to  one-half  the  elastic 
limit,  what  must  be  the  width  of  a piece  of  bar  steel  (structural) 
to  carry  a load  cf  50000  pounds,  if  the  thickness  of  bar  is  ?>/■'"  ~ 

4.  A bar  of  structural  steel  1 c/4,!  in  diameter  fails  under 
a tensile  load  cf  150000  pounds,  what  was  the  ultimate  tensile 
strength  of  bar? 

5.  Using  same  tensile  strength  as  in  problem  {-'■)  what 
will  be  the  unit  breaking  stress  in  a round  bar  of  structural 
steel  1 7/bn  diameter? 

6.  A steel  bar  1,;  in  diameter  and  10,f  long  has  a total 
value  for  the  elastic  limit  cf  30000  pounds,  and  a tot"!  ultimate 
tensile  strength  of  50000  pounds.  The  length  of  the  bar  at  the 
elastic  limit  was  10.0180"  and  at  failure  was  15. "8".  Find 
elastic  limit,  ultimate  tensile  strength,  and  the  unit  elongation 
for  the  elastic  limit  and  for  failure. 

7.  The  maximum  pressure  in  a steam  engine  cylinder,  l°ff 
diameter,  is  1 20  pounds  per  square  inch.  Using  a factor  of 
safety  of  10  find  diameter  of  piston  rod. 

8.  A short  (structural)  steel  bar  is  3"  in  diameter,  -;rhat 

cempressiwe  load  will  it  take  using  a factor  of  safety  cf  3?  cf  ic 

ISC 


' 


2h 


* 

• 

* 

I 


. 


. 


■ 


■ 


•j 

t!  - < ? 

: 

, • 

. 


. 


. 


1 on 

-L  / 


9,  Find  the  load  which  a short,  hollow  coot  iron  coliwn 
IF"  external  dia:  eter , 9 1/2"  internal  diameter,  wil.l  take  with 

factor  of  safety  of  6. 

10.  A simple  beam  is  58"  long  and  carries  a concentrated 
load  of  6500  pounds  at  a distance  17"  toward  the  right  from, 
the  left  support . Find  both  reactions,  How  many  forces  acting 
on  the  beams.  Make  a sketch  indicating  direction  and  magnitude 

of  each  force. 

11.  A simple  beam  OF"  long  carries  two  cone  n t rated  lea's 
of  5350  and  1975  pounds.  The  two  loads  are  1C"  apart  and  the 
former  is  15"  from  the  left  support.  Find  both  reactions. 

How  many  forces  acting  on  the  beam?  Mak'  sketch  indicating 
direction  and.  magnitude  of  each  force. 

IF.  A simple  beam  26"  between,  sun  -orts  earnin'-  three  con- 
centrated loads  of  1800 , 5000  and  lF0n  pounds.  The  first  is 
7"  to  the  left  of  the  left  support , the  second  to  the  right 
of  the  left  supnort , and  the  third  is  9"  to  the  left  of  the 
right  support.  Find  both  reactions.  Make  sketch  indicating 
direction  and  magnitude  of  each  force. 

15.  A simple  beam  2F"  between  sup sorts  carries  a load  of 
3-050  pounds  at  a distance  of  5"  to  the  left  of  the  left  sun  ert. 
Find  reactions.  Make  sketch  indicating  direction  an-’  magnitude 
of  eash  force. 

14.  In  protier’s  ( 10) , (ll),  (IF)  and  (13)  find 

(a ) Bending  moment  in  the  plane  of  each  fo^ce. 

(b)  Draw  diagram  of  bending  moment  due  to  each  load, 

(a)  " " " " " " " total  loads, 

192 


- 

-V 

' 


(d)  Locate  section  in  bean  of  maximum  bonding  moment . 

(e)  Find  size  of  bean  assuming  circular  section  and 


St  = 9000  pounds  per  square  inch. 

15 o A gear  wheel  SO”  in  diameter  has  5 arms  and  tr.  nsraito 
18  E.P.  at  800  R.P.M. 

(a)  What  kind  of  beams  are  the  arms? 

i 

(b)  Find  bending  moment  which  each  arm  must  resist. 

(c)  Find  size  of  arms  at  center  of  gear  assuming 
elliptical  section  with  major  axis  equal  to  twice  the  minor  axis. 
Take  St  = 2000  pounds  per  square  inch.  I for  ellipse 

16.  In  problem,  (ll)  assume  rectangular  section  with  depth- 
equal  to  three  tim.es  the  breadth.  Find  size  of  beam.  Take  St  = 800 


pound::-  per  square  inch. 

17.  In  a single  riveted  lap  joint  the  thickness  of  the 
plate  is  3/3"  , diameter  of  rivets  p/d"  , pitch  of  rivets  2 P/l6t?  , 
margin  1 l/d"  , Sketch  joint  (two  views)  and  find  the  fo  1.1  owing*. 


(a 

) Resistance 

of 

the. 

joint 

to 

she  ring  of  rivets. 

(b 

\ fr 

/ 

it 

11 

ir 

crushing  the  plate 

and  rivets. 

(0 

) 

11 

11 

11 

it 

bursting  of  plate 

opposite  the 

rivet . 

(d 

} Resistance 

11 

u 

ii 

tt 

tearing  apart  of  plate 

(as  shown  at 

CL  Fig.  9 ) , 

(e 

) Resistance 

ii 

11 

11 

11 

s h e a r i n g of  > 1 a t e , 

. 


. 


r • , 


. 


. 

] °9 

18.  A double  riveted  lap  joint  has  t’  o following  dimen- 
sions : Thickness  of  plo.ro  1 /?"  , diameter  of  rivets  7 j 8”  , 

pitch  of  rivets  1 13/l6”  , margin  1 7 /.1G”  . Sketch  joint  and 
investigate  as  in  problem  (17)  using  the  following  working 
stresses.  - St  - -8000,  Ss  = G300  and  = 130n0  pounds  per 
sq.  in, 

19.  A triple  riveted  butt  joint  has  the  following  dimen- 
sions: Thickness  of  plates  5/3” , diameter  of  rivets  1”,  thick- 
ness of  strops  l/s" , width  of  outer  plate  11  5/8” , width  cf 

inner  plate  18”,  pitch  of  rivets  5 7/8”,  and  7 3/4”  ( -outside 
rovrs),  distance  apart  of  rivet  rows  3 3/lo”  and  2 5/8”  (two 
rows  on  each  side  of  joint).  Sketch  joint  and  investigate 
as  in  problem  (17)  using  the  following  - St  = 75nr',  S3  = 5000 
and  S = 10000  pounds  per  square  inch. 

20  n The  outside  diameter  of  a bolt  threaded.  U.S.Std.  is 
1 3/d” . Find  pitch,  number  of  threads  per  inch  and.  tensile 
strength  (St  = 4000  pounds  per  square  inch). 

81.  Find  diameter  of  wrought  iron  holt  that  is  to  sustain 
a ste:dy  load  of  5 tons. 

22,  VJliat  steady  load  v.ill  7 studs  bolts  (U.S.Std.)  S/'” 
in  diameter  safely  resist? 

33.  Calculate  diameter  of  wrought  iron  bo], t that  is  to 

sustain  a varying  load  of  1800  pounds. 

84.  Sketch  and.  dimension  fully  a finished  bolt  7/8" 

diameter  with  hexagonal  head  and.  nut. 

193 


* 


■ i<t 

■ 

■ 

■ 

130 


25,  In  a bolt  1”  diameter  (V  threads ) calculate  height  of 
head  in  order  that  the  latter  may  bo  as  strong  to  resist  failure 
by  shearing  as  the  bolt  is  to  resist  failure  by  tearing  apart, 

26,  Same  as  problem  25  except  U * S.  Std.  threads, 

27,  Calculate  diameter  and  number  of  stud  bolts  required 
to  secure  cylinder  head  to  cylinder,  the  diameter  of  "'rich  is 

12” , maximum  steam  pressure  120  pounds  per  square  inch.  (Take 
St  = 4500  pounds  per  square  inch, ) 

2G » ".'hat  force  must  a workman  exert  at  the  e*  d of  a wrench 
10-n  in  length  in  order  to  cause  failure  of  a 5/b"  diameter  bolt 
(U.3.  Std.  thread)  (a)  by  stripping  of  threads  (b)  by  pulling 
apart  of  bolt, 

29.  In  problem  (27)  find  the  depth  into  cylinder  in  which, 
stud  bolts  should  ho  screwed  in  order  that  in  cast  iron  should 

o 

not  exceed  1800  pounds  per  square  inch, 

30.  Draw  full  size  a finished  bolt  5"  long  with  square 
head  and  hexagonal  nut.  Bolt  is  to  be  subjected  to  a varying  load 

cf  3500  pounds, 

•rl . A flat  key  in  a c"  diameter  shaft  has  the  fo]  lowing 

dimensions:  7/l6"  x 5/4"  Y 6".  Find 

(a)  Resistance  of  ke^v  to  shearing. 

(b)  ” 11  " ,f  crushing. 

Take  Sg  = 7500  and  SQ  - 15000  pounds  per  square  inch. 

32.  Y.'hat  horse  power  wi  1/the  key  in  problem  (31) 
safely  transmit  at  200  R.  P.  M. ? 

192 


■ 


■ 


‘ 


. 


t 


131 


35.  A 4"  shaft  is  to  transmit  250  H.Po  at  300  E.P.h. 

If  key  is  8"  long  and  of  square  cross  section  determine  dimensions 

of  key. 

34„  If  a 3 l/2"  diameter  of  shaft  is  to  transmit  100  HcP, 
at  100  RoPcMc  and  the  ratio  of  depth  of  key  to  breadth  is  as  2 
to  3 find  size  of  key.  Take  length  of  key  equal  to  7",  also 
Ss  = 6000  and  SG  = 12000  pounds  per  sq»  in. 

35.  A lever  30"  long  is  keyed  to  a brake  shaft  1 1 / 4"  in 
diameter.  If  maximum  load  at  end  of  lever  is  200  pounds  and 
width  of  key  is  7/l6"  what  should  be  the  length  in  order  that 
Ss  may  not  exceed  8000  pounds  per  square  inch. 

36.  If  in  problem  (35)  the  key  is  square  in  cross  section 
determine  the  unit  crushing  stress. 

37.  Referring  to  Fig.  23  if  d = 2"  calculate  d,  t,  t,  e, 
a,  and  D for  equal  strength  with  rod.  (St  = 6000,  Ss  = 4500  and 
S0  = 12000  ) c 

38.  A cotter  l/2,f  thick  and  2 l/2"  deep  must  resist  a load 
taken  by  a 2"  bolt,  the  maximum  unit  stress  in  which  is  10000  pounds 
per  square  inch.  If  cotter  is  1"  from  end  of  bolt  what  is  value 

of  unit  shearing  stress  developed  in  both  cotter  and  bolt9  ^lso 
what  is  unit  bearing  value  of  bolt  against  cotter? 

39.  The  end  of  a 3"  round  rod  is  enlarged  to  3 ll/l0"  dia- 
meter and  is  fitted  with  a cotter  lc/l6"  thick  whose  denth  is 

3 15/16"  . Using  same  stresses  as,,  in  problem  (37)  determine  the 
weakest  part  of  joint. 

192 


. 


■ 


. 


.. 

1 1 


152 


40.  In  problem  (57)  find  diameter  of  outer  rod  above,  also 

below  cotter. 

41 o The  driving  pulley  of  a machine  in  2C"  in  diameter 
and  the  effective  pull  on  belt  is  ISO  pounds,  Y.'hat  H.P.  is  given 
to  machine  when  pulley  makes  125  EBP.M,? 

42 . A cold  rolled  steel  shaft  10“  between  bearings  transmits 
75  H.P.  at  200  R.P.M.  Find 

(a)  Maximum  bending  moment  in  shaft 

(b)  Twisting  moment  in  shaft. 

4-5.  Find  diameter  of  shaft  required  ( 3 p = 10000  and  Ss  ~ 

8000  pounds  per  sq.  in.) 

(a)  To  resist,  bending  moment  in  problem  .40. 

(b)  " " twisting  “ " “ 4°. 

44.  Find  diameter  of  shaft  required  to  transmit  both  the 
bending  moment  and  the  twisting  moment  simultaneously  as  given  in 
problem  40.  (S^.  = 10000  and  = 80On  pounds  per  sq . in.) 

45.  A shaft  whose  bearings  are  21"  apart  has  keyed  to  tt 
two  gear  wheels , 9“  and  21  l/g"  pitch  di  meter.  The  former  is 

0‘*  to  the  right  of  the  left  bearing  and  the  latter  7"  to  tn©  left 
of  the  right  bearing.  The  shaft  transmits  100  H.P.  at  20  R.P.n., 
the  power  being  delivered  to  gear  in  a horizontal  line  above  shaft 
and  delivered  from  pinion  in  a horizontal  lire  belovr  shaft. 

(a)  Sketch  arrangement  indicating  magnitude  and  direc- 
tion of  all  forces  acting  on  shaft. 

(b)  Find  diameter  of  shaft  using  same  working  stresses 
as  in  problem  45. 


192 


■ V 

' 1 • 

• 

, 


\ 


153 


46.  DFind  angle  of  twist  in  problem  4-5.  (Angle  0.  See  Fig, 

85) . 

47«  Referring  to  Fig.  26  , CB  = 23",  AD  ~ 10"  and  CB  = ll'1  « 

The  pitch  diameter  6f  gears  e_  and  f are  19"  and  8 3/4"  resnoctively 
and  the  direction  of  pressures  against  gear  teeth  are  as  indicated 
in  Fig.  26.  If  shaft  is  to  transmit  275  H.P.  at  995  find 

diameter  of  shaft.  Take  = 12000  and  Sg  = 100O0  pounds  per 
sq,  in, 

48.  In  Fig.  27,  AD  = C5  = 7"  and  AC  = 12",  also  diameter 
of  gear  0 = 17"  and  diameter  of  gear  D = diameter  of  gear  E = 8" . 

If  3 haft  is  to  transmit  50  H«P.  at  60  R.P.Ii.  and  one-half  the 
power  is  taken  off  at  gear  D and  the  other  half  at  gear  E,  find 
diameter  of  shaft.  Use  same  working  stresses  as  in  problem  47. 

Take  pressures  against  gear  teeth  as  indicated  in  Fig.  27. 

49 o A shaft  15"  in  diameter  is  fitted  with  a crank  of  radius 
40" c The  maximum  thrust  on  crank  end  is  149000  pounds , what  is  the 
fiber  stress  in  shaft  due  to  twisting  moment? 

50.  The  outer  siameter  of  a hollow  shaft  is  two  tiros  the  in- 
ner diameter.  Keyed  to  this  shaft  is  a crank  90"  in  length.  If 
pressure  against  crank  pin  is  150000  pounds,  find  diameter  (outside) 
of  shaft.  Take  Sg  = 8000  pounds  per  sq.  in. 

51 « What  size  of  journals  are  required  in  problem  (4-2)  assum- 
ing pressure  per  square  inch  of  projected  area  of  bearing  = 950 
pounds  and  length  of  journal  = 5 times  the  diameter  of  journal? 

52 « Check  size  of  journals  obtained  in  problem  (51)  for  i oth 
strength  and  stiffness. 

192 


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530  Refer  to  problem  (-1-7)  and  compute  size  of  journals, 
also  thickness  of  cast  iron  caps  and  die motor  of  wrought  iron 
cap  bolts  (p=  700  and  1=2  l/7d). 

54o  An  overhanging  shaft  has  keyed  to  it  a spur  gear  12” 
pitch  diameter,  7”  to  the  loft  of  the  left  bearing,,  The  distance 
between  the  two  bearings  is  16”.  If  gear  transmits  70  K ,P . at 
125  R.P.Ii.l  find 

(a)  Dimensions  of  journals  (p  = 750  and  1=2  l/2d) 

(b)  Thickness  of  cast  iron  caps. 

(c)  Diameter  of  wrought  iron  cap  bolts. 

55.  The  main  front  bearing  of  a punching  machine  must 
take  a thrust  of  106000  pounds.  Required  dimensions  of  journal 
assuming  p = 2500  and  1 = 5 d « 

56.  The  distance  between  two  parallel  shafts,  connected  by 
spur  gears  is  22  l/2”  and  the  velocity  ratio  of  the  gears  is  Z/Z» 

F i nd  di ame ter  of  gears . 

(Note:  In  rough  oast  spurs  gears  take  f = 2 l/2p*  (ammor.- 
imatoly)  arxl  in.  cut  spur  gears  take  f = ( approximately ) ) . 

57.  In  a pair  of  cast  iron  spur  gears,  with  cut  teeth,  the 
diameter  of  the  driver  is  0 l/2”  and  its  R.P.ii,  is  150.  If 
velocity  ratio  is  5/2  and  pressure  between  gears  is  150  pounds, 
find 

(a)  Diameter  of  driven  gear. 

(b)  Distance  between  two  shafts. 

(o)  Horse  power  transmitted. 

192 


' 


58. 


A oast  iron  spur  gear  with  rough  oust  to'-th,  9 l/d" 
pitch,  SIT,  running  at  10°  R.P.M.  transmits  what  horse  power? 

59.  A rough  oast  spur  gear  of  cast  iron,  1 5/4"  pitch, 

14.52"  P.D,,  5"  F transmits  what  horse  power  at  125  R.P.M,? 

60.  In  a pair  of  cast  steel  spur  gears,  with  cast  teeth  the 
diameter  of  the  driver  is  18  3/4".  If  the  velocity  ratio  is 

3 to  1 and  H.P.  transmitted  is  250  when  R.P.Ii.  of  driver  is 
180  find 

(a)  Distance  between  shafts. 

(b)  Face  of  gears  (e)  Dedendum 

(c)  Circular  pitch  (f)  Thickness  of  tooth  on  pitch  line 

(d)  Addendum  (g)  Diameter  of  base  circle. 

61.  A cast  iron  spur  gear  with  cut  teeth  3p,  48"  p.d,,  3 l/p" 
f.  transmits  what  H.P.  at  150  R.P.M. ‘3 

62.  A pair  of  cast  iron  spur  gears  with  cut  teeth  is 
required  to  transmit  15  H.P.  .The  distance  between  two  shafts  is 
14  5/4"  and  the  velocity  ratio  is  12/7,  If  the  diameter  of 
driver  is  11  l/2tr  and  its  R.P.M.  is  180,  find 

( a ) D i ame t er  and  R . P . M . of  driven 

(b)  Diametral  pitch,  and  number  of  cutter  for  each 

gear. 

63.  In  problem  i'62)  if  the  distance  between  bearings  is  21, 

find 

(a)  Diameter  of  each  shaft.  (C.F.S.) 

(b)  Size  cf  journals.  (p  = 500  and  1 = 3d) 

(c)  Thickness  of  cast  cap3. 

(d)  Diameter  of  17. E.  cap  bolts 0 

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64.  In  problem  62,  find  size  of  arms  of  gear,  (assume 
elliptical  cross  section  with  major  axis  = twice  minor  axis  also 
6 arms).  Take  = 2500. 

65 e In  a pair  of  spur  gears  with  cut  teeth  the  pinion  is 
made  of  machinery  steel  and  the  gear  of  cast  iron.  The  diameter 
of  driver  is  5"  and  the  velocity  ratio  is  8.2  to  1.  find  diame- 
tral pitch,  and  number  of  cutter  required  in  each  case. 


66 . 

In  problem  65  find  ratio  of  strength  of  pinion  to 

strength 

of  gear. 

67. 

A cast  iron  bevel  gear  with  cast  teeth,  1 1 nitch, 

60T  will 

transmit  what  H.P.  at  150  R.P.M. ? 

68. 

In  problem  (67)  find  formative  number  of  teeth,  (center 

angle  = 32°). 

69.  Show  that  formative,  number  of  teeth  is  equal  to  actual 
number  of  teeth  in  bevel  gear  multiplied  by  the  ratio  — , (See  Fig. 


52  ) , 

r 

70. 

In  a pair  of  cast  iron  bevel  gears  with  cast  teeth; 

the  diameter  of  pinion' is  7.55'1  and  the  diameter  of  gear  is  25,79". 
What  is  velocity  ratio  if  pinion  is  driver? 


71 , 

In  problem  70  if  the  geah  is  driver  and  its  R.P.M.  is 

200  find 

H.P.  transmitted.  Take  pitch  ~ 1 l/p" . 

72, 

Find  circular  pitch  of  bevel  gear  °9,61lf  diameter, 

which  must  transmit  30  H.P . at  150  R.P.M. 

73.  Design  a pair  of  cast  iron  bevel  gears  to  trasnnit  55 
E.P..  R.P.M.  of  driver  = 60;  velocity  ratio  = 3/7;  diameter  f 
pinion  = 15"  . Teeth  to  be  standard  involute  cut  on  a milling 


machine « 


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THE  U*  SARY 


Number  or  Teeth. 


0 10  EO  30  -40  SO  &0  70  80  90  IOO  II O IZO  \ 30  140 

Number  of  Te®t-h 


Fig  41 


Fig  42 


Values  of  Strength  Foic.+or 


p 


Fig  27  and  28 


Fig  29 


Fig  45 


THE  UBKAWY 
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UMYE&Slflf  GF  IU-UI01S 


Fig  50 


THE  LIBRARY 
OF  THE 

UNIVERSITY  Cr  ILLINOIS 


IKE  U2»  r< 

APR  2 4 1930 

UNIVERSITY  of  * 


